{
  "index": "1941-A-5",
  "type": "GEO",
  "tag": [
    "GEO",
    "ALG"
  ],
  "difficulty": "",
  "question": "5. Show that the line which moves parallel to the plane \\( y=z \\) and which intersects the two parabolas \\( y^{2}=2 x, z=0 \\) and \\( z^{2}=3 x, y=0 \\) sweeps out the surface\n\\[\nx=(y-z)\\left(\\frac{y}{2}-\\frac{z}{3}\\right) .\n\\]",
  "solution": "Solution. Suppose that the line meets the first parabola at \\( \\left(18 s^{2}, 6 s, 0\\right) \\) and the second parabola at a distinct point \\( \\left(12 t^{2}, 0,6 t\\right) \\). The condition that the line be parallel to the plane \\( y=z \\) is that the direction vector of the line, namely\n\\[\n\\left(18 s^{2}-12 t^{2}, 6 s,-6 t\\right)\n\\]\nshould lie in that plane, i.e., \\( 6 s=-6 t \\). Hence the two points are \\( \\left(18 t^{2}\\right. \\), \\( -6 t, 0) \\) and \\( \\left(12 t^{2}, 0,6 t\\right) \\), where \\( t \\neq 0 \\).\n\nThis line has the parametric representation\n\\[\n\\left(18 t^{2},-6 t, 0\\right)+u\\left(6 t^{2},-6 t,-6 t\\right)\n\\]\nwhere \\( u \\) is the parameter. So we have the two-parameter representation\n\\[\n\\begin{array}{l}\nx=18 t^{2}+6 u t^{2} \\\\\ny=-6 t-6 u t \\\\\nz=-6 t u\n\\end{array}\n\\]\nwhere \\( t \\neq 0 \\), for the surface generated by the moving line.\nWe eliminate the parameters by noting that\n\\[\ny-z=-6 t, \\quad \\text { and }\n\\]\n\\[\n\\frac{y}{2}-\\frac{z}{3}=-t(3+u) .\n\\]\n\nSo we get\n\\[\nx=6 t(3 t+u t)=(y-z)\\left(\\frac{y}{2}-\\frac{z}{3}\\right) .\n\\]\n\nThis proves that any point on any line parallel to the plane \\( y=z \\) which meets the two given parabolas at distinct points lies on the surface (3).\n\nConversely, if \\( \\left(x_{1}, y_{1}, z_{1}\\right) \\) is a point on surface (3), with \\( y_{1} \\neq z_{1} \\), the numbers \\( t \\) and \\( u \\) can be determined from (2) so that \\( \\left(x_{1}, y_{1}, z_{1}\\right) \\) has the form (1). Points on the plane \\( y=z \\), however, are special because the two given parabolas intersect the plane at the point \\( (0,0,0) \\). So every point of the plane is on some line which is parallel to (in fact contained in) the plane and which intersects both parabolas. If these lines are to be taken into account, then the locus (3) must be supplemented by adjoining the whole plane \\( y=z \\). The surface (3) itself meets the plane \\( y=z \\) in just one line, \\( x=0, y=z \\).",
  "vars": [
    "x",
    "y",
    "z",
    "x_1",
    "y_1",
    "z_1"
  ],
  "params": [
    "s",
    "t",
    "u"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "coordx",
        "y": "coordy",
        "z": "coordz",
        "x_1": "pointx",
        "y_1": "pointy",
        "z_1": "pointz",
        "s": "sliding",
        "t": "variable",
        "u": "parameter"
      },
      "question": "5. Show that the line which moves parallel to the plane \\( coordy=coordz \\) and which intersects the two parabolas \\( coordy^{2}=2 coordx, coordz=0 \\) and \\( coordz^{2}=3 coordx, coordy=0 \\) sweeps out the surface\n\\[\ncoordx=(coordy-coordz)\\left(\\frac{coordy}{2}-\\frac{coordz}{3}\\right) .\n\\]",
      "solution": "Solution. Suppose that the line meets the first parabola at \\( \\left(18 sliding^{2}, 6 sliding, 0\\right) \\) and the second parabola at a distinct point \\( \\left(12 variable^{2}, 0,6 variable\\right) \\). The condition that the line be parallel to the plane \\( coordy=coordz \\) is that the direction vector of the line, namely\n\\[\n\\left(18 sliding^{2}-12 variable^{2}, 6 sliding,-6 variable\\right)\n\\]\nshould lie in that plane, i.e., \\( 6 sliding=-6 variable \\). Hence the two points are \\( \\left(18 variable^{2}\\right. \\), \\( -6 variable, 0) \\) and \\( \\left(12 variable^{2}, 0,6 variable\\right) \\), where \\( variable \\neq 0 \\).\n\nThis line has the parametric representation\n\\[\n\\left(18 variable^{2},-6 variable, 0\\right)+parameter\\left(6 variable^{2},-6 variable,-6 variable\\right)\n\\]\nwhere \\( parameter \\) is the parameter. So we have the two-parameter representation\n\\[\n\\begin{array}{l}\ncoordx=18 variable^{2}+6 parameter variable^{2} \\\\\ncoordy=-6 variable-6 parameter variable \\\\\ncoordz=-6 variable parameter\n\\end{array}\n\\]\nwhere \\( variable \\neq 0 \\), for the surface generated by the moving line.\nWe eliminate the parameters by noting that\n\\[\ncoordy-coordz=-6 variable, \\quad \\text { and }\n\\]\n\\[\n\\frac{coordy}{2}-\\frac{coordz}{3}=- variable(3+parameter) .\n\\]\n\nSo we get\n\\[\ncoordx=6 variable(3 variable+parameter variable)=(coordy-coordz)\\left(\\frac{coordy}{2}-\\frac{coordz}{3}\\right) .\n\\]\n\nThis proves that any point on any line parallel to the plane \\( coordy=coordz \\) which meets the two given parabolas at distinct points lies on the surface (3).\n\nConversely, if \\( \\left(pointx, pointy, pointz\\right) \\) is a point on surface (3), with \\( pointy \\neq pointz \\), the numbers \\( variable \\) and \\( parameter \\) can be determined from (2) so that \\( \\left(pointx, pointy, pointz\\right) \\) has the form (1). Points on the plane \\( coordy=coordz \\), however, are special because the two given parabolas intersect the plane at the point \\( (0,0,0) \\). So every point of the plane is on some line which is parallel to (in fact contained in) the plane and which intersects both parabolas. If these lines are to be taken into account, then the locus (3) must be supplemented by adjoining the whole plane \\( coordy=coordz \\). The surface (3) itself meets the plane \\( coordy=coordz \\) in just one line, \\( coordx=0, coordy=coordz \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "meadowfox",
        "y": "lanterncat",
        "z": "raspberry",
        "x_1": "silenthawk",
        "y_1": "thunderpig",
        "z_1": "oceanmoss",
        "s": "copperant",
        "t": "velvetlion",
        "u": "marblehare"
      },
      "question": "5. Show that the line which moves parallel to the plane \\( lanterncat=raspberry \\) and which intersects the two parabolas \\( lanterncat^{2}=2 meadowfox, raspberry=0 \\) and \\( raspberry^{2}=3 meadowfox, lanterncat=0 \\) sweeps out the surface\n\\[\nmeadowfox=(lanterncat-raspberry)\\left(\\frac{lanterncat}{2}-\\frac{raspberry}{3}\\right) .\n\\]\n",
      "solution": "Solution. Suppose that the line meets the first parabola at \\( \\left(18 copperant^{2}, 6 copperant, 0\\right) \\) and the second parabola at a distinct point \\( \\left(12 velvetlion^{2}, 0,6 velvetlion\\right) \\). The condition that the line be parallel to the plane \\( lanterncat=raspberry \\) is that the direction vector of the line, namely\n\\[\n\\left(18 copperant^{2}-12 velvetlion^{2}, 6 copperant,-6 velvetlion\\right)\n\\]\nshould lie in that plane, i.e., \\( 6 copperant=-6 velvetlion \\). Hence the two points are \\( \\left(18 velvetlion^{2}\\right. \\), \\( -6 velvetlion, 0) \\) and \\( \\left(12 velvetlion^{2}, 0,6 velvetlion\\right) \\), where \\( velvetlion \\neq 0 \\).\n\nThis line has the parametric representation\n\\[\n\\left(18 velvetlion^{2},-6 velvetlion, 0\\right)+marblehare\\left(6 velvetlion^{2},-6 velvetlion,-6 velvetlion\\right)\n\\]\nwhere \\( marblehare \\) is the parameter. So we have the two-parameter representation\n\\[\n\\begin{array}{l}\nmeadowfox=18 velvetlion^{2}+6 marblehare velvetlion^{2} \\\\\nlanterncat=-6 velvetlion-6 marblehare velvetlion \\\\\nraspberry=-6 velvetlion marblehare\n\\end{array}\n\\]\nwhere \\( velvetlion \\neq 0 \\), for the surface generated by the moving line.\nWe eliminate the parameters by noting that\n\\[\nlanterncat-raspberry=-6 velvetlion, \\quad \\text { and }\n\\]\n\\[\n\\frac{lanterncat}{2}-\\frac{raspberry}{3}=-velvetlion(3+marblehare) .\n\\]\n\nSo we get\n\\[\nmeadowfox=6 velvetlion(3 velvetlion+marblehare velvetlion)=(lanterncat-raspberry)\\left(\\frac{lanterncat}{2}-\\frac{raspberry}{3}\\right) .\n\\]\n\nThis proves that any point on any line parallel to the plane \\( lanterncat=raspberry \\) which meets the two given parabolas at distinct points lies on the surface (3).\n\nConversely, if \\( \\left(silenthawk, thunderpig, oceanmoss\\right) \\) is a point on surface (3), with \\( thunderpig \\neq oceanmoss \\), the numbers \\( velvetlion \\) and \\( marblehare \\) can be determined from (2) so that \\( \\left(silenthawk, thunderpig, oceanmoss\\right) \\) has the form (1). Points on the plane \\( lanterncat=raspberry \\), however, are special because the two given parabolas intersect the plane at the point \\( (0,0,0) \\). So every point of the plane is on some line which is parallel to (in fact contained in) the plane and which intersects both parabolas. If these lines are to be taken into account, then the locus (3) must be supplemented by adjoining the whole plane \\( lanterncat=raspberry \\). The surface (3) itself meets the plane \\( lanterncat=raspberry \\) in just one line, \\( meadowfox=0, lanterncat=raspberry \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "verticalaxis",
        "y": "horizontalaxis",
        "z": "surfacelayer",
        "x_1": "verticalaxispivot",
        "y_1": "horizontalaxispivot",
        "z_1": "surfacelayerpivot",
        "s": "endpoint",
        "t": "timelessness",
        "u": "stagnation"
      },
      "question": "5. Show that the line which moves parallel to the plane \\( horizontalaxis=surfacelayer \\) and which intersects the two parabolas \\( horizontalaxis^{2}=2 verticalaxis, surfacelayer=0 \\) and \\( surfacelayer^{2}=3 verticalaxis, horizontalaxis=0 \\) sweeps out the surface\n\\[\nverticalaxis=(horizontalaxis-surfacelayer)\\left(\\frac{horizontalaxis}{2}-\\frac{surfacelayer}{3}\\right) .\n\\]",
      "solution": "Solution. Suppose that the line meets the first parabola at \\( \\left(18 endpoint^{2}, 6 endpoint, 0\\right) \\) and the second parabola at a distinct point \\( \\left(12 timelessness^{2}, 0,6 timelessness\\right) \\). The condition that the line be parallel to the plane \\( horizontalaxis=surfacelayer \\) is that the direction vector of the line, namely\n\\[\n\\left(18 endpoint^{2}-12 timelessness^{2}, 6 endpoint,-6 timelessness\\right)\n\\]\nshould lie in that plane, i.e., \\( 6 endpoint=-6 timelessness \\). Hence the two points are \\( \\left(18 timelessness^{2}\\right. , -6 timelessness, 0) \\) and \\( \\left(12 timelessness^{2}, 0,6 timelessness\\right) \\), where \\( timelessness \\neq 0 \\).\n\nThis line has the parametric representation\n\\[\n\\left(18 timelessness^{2},-6 timelessness, 0\\right)+stagnation\\left(6 timelessness^{2},-6 timelessness,-6 timelessness\\right)\n\\]\nwhere \\( stagnation \\) is the parameter. So we have the two-parameter representation\n\\[\n\\begin{array}{l}\nverticalaxis=18 timelessness^{2}+6 stagnation timelessness^{2} \\\\\nhorizontalaxis=-6 timelessness-6 stagnation timelessness \\\\\nsurfacelayer=-6 timelessness stagnation\n\\end{array}\n\\]\nwhere \\( timelessness \\neq 0 \\), for the surface generated by the moving line.\nWe eliminate the parameters by noting that\n\\[\nhorizontalaxis-surfacelayer=-6 timelessness, \\quad \\text { and }\n\\]\n\\[\n\\frac{horizontalaxis}{2}-\\frac{surfacelayer}{3}=-timelessness(3+stagnation) .\n\\]\n\nSo we get\n\\[\nverticalaxis=6 timelessness(3 timelessness+stagnation timelessness)=(horizontalaxis-surfacelayer)\\left(\\frac{horizontalaxis}{2}-\\frac{surfacelayer}{3}\\right) .\n\\]\n\nThis proves that any point on any line parallel to the plane \\( horizontalaxis=surfacelayer \\) which meets the two given parabolas at distinct points lies on the surface (3).\n\nConversely, if \\( \\left(verticalaxispivot, horizontalaxispivot, surfacelayerpivot\\right) \\) is a point on surface (3), with \\( horizontalaxispivot \\neq surfacelayerpivot \\), the numbers \\( timelessness \\) and \\( stagnation \\) can be determined from (2) so that \\( \\left(verticalaxispivot, horizontalaxispivot, surfacelayerpivot\\right) \\) has the form (1). Points on the plane \\( horizontalaxis=surfacelayer \\), however, are special because the two given parabolas intersect the plane at the point \\( (0,0,0) \\). So every point of the plane is on some line which is parallel to (in fact contained in) the plane and which intersects both parabolas. If these lines are to be taken into account, then the locus (3) must be supplemented by adjoining the whole plane \\( horizontalaxis=surfacelayer \\). The surface (3) itself meets the plane \\( horizontalaxis=surfacelayer \\) in just one line, \\( verticalaxis=0, horizontalaxis=surfacelayer \\)."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "y": "hjgrksla",
        "z": "mvdprqoe",
        "x_1": "lqnfsmda",
        "y_1": "swvpkjrh",
        "z_1": "cegxdnoh",
        "s": "atosirle",
        "t": "jrvumkeh",
        "u": "fqdwnpsi"
      },
      "question": "5. Show that the line which moves parallel to the plane \\( hjgrksla=mvdprqoe \\) and which intersects the two parabolas \\( hjgrksla^{2}=2 qzxwvtnp, mvdprqoe=0 \\) and \\( mvdprqoe^{2}=3 qzxwvtnp, hjgrksla=0 \\) sweeps out the surface\n\\[\nqzxwvtnp=(hjgrksla-mvdprqoe)\\left(\\frac{hjgrksla}{2}-\\frac{mvdprqoe}{3}\\right) .\n\\]",
      "solution": "Solution. Suppose that the line meets the first parabola at \\( \\left(18 atosirle^{2}, 6 atosirle, 0\\right) \\) and the second parabola at a distinct point \\( \\left(12 jrvumkeh^{2}, 0,6 jrvumkeh\\right) \\). The condition that the line be parallel to the plane \\( hjgrksla=mvdprqoe \\) is that the direction vector of the line, namely\n\\[\n\\left(18 atosirle^{2}-12 jrvumkeh^{2}, 6 atosirle,-6 jrvumkeh\\right)\n\\]\nshould lie in that plane, i.e., \\( 6 atosirle=-6 jrvumkeh \\). Hence the two points are \\( \\left(18 jrvumkeh^{2}\\right. , -6 jrvumkeh, 0) \\) and \\( \\left(12 jrvumkeh^{2}, 0,6 jrvumkeh\\right) \\), where \\( jrvumkeh \\neq 0 \\).\n\nThis line has the parametric representation\n\\[\n\\left(18 jrvumkeh^{2},-6 jrvumkeh, 0\\right)+fqdwnpsi\\left(6 jrvumkeh^{2},-6 jrvumkeh,-6 jrvumkeh\\right)\n\\]\nwhere \\( fqdwnpsi \\) is the parameter. So we have the two-parameter representation\n\\[\n\\begin{array}{l}\nqzxwvtnp=18 jrvumkeh^{2}+6 fqdwnpsi jrvumkeh^{2} \\\\\nhjgrksla=-6 jrvumkeh-6 fqdwnpsi jrvumkeh \\\\\nmvdprqoe=-6 jrvumkeh fqdwnpsi\n\\end{array}\n\\]\nwhere \\( jrvumkeh \\neq 0 \\), for the surface generated by the moving line.\nWe eliminate the parameters by noting that\n\\[\nhjgrksla-mvdprqoe=-6 jrvumkeh, \\quad \\text { and }\n\\]\n\\[\n\\frac{hjgrksla}{2}-\\frac{mvdprqoe}{3}=-jrvumkeh(3+fqdwnpsi) .\n\\]\n\nSo we get\n\\[\nqzxwvtnp=6 jrvumkeh(3 jrvumkeh+fqdwnpsi jrvumkeh)=(hjgrksla-mvdprqoe)\\left(\\frac{hjgrksla}{2}-\\frac{mvdprqoe}{3}\\right) .\n\\]\n\nThis proves that any point on any line parallel to the plane \\( hjgrksla=mvdprqoe \\) which meets the two given parabolas at distinct points lies on the surface (3).\n\nConversely, if \\( \\left(lqnfsmda, swvpkjrh, cegxdnoh\\right) \\) is a point on surface (3), with \\( swvpkjrh \\neq cegxdnoh \\), the numbers \\( jrvumkeh \\) and \\( fqdwnpsi \\) can be determined from (2) so that \\( \\left(lqnfsmda, swvpkjrh, cegxdnoh\\right) \\) has the form (1). Points on the plane \\( hjgrksla=mvdprqoe \\), however, are special because the two given parabolas intersect the plane at the point \\( (0,0,0) \\). So every point of the plane is on some line which is parallel to (in fact contained in) the plane and which intersects both parabolas. If these lines are to be taken into account, then the locus (3) must be supplemented by adjoining the whole plane \\( hjgrksla=mvdprqoe \\). The surface (3) itself meets the plane \\( hjgrksla=mvdprqoe \\) in just one line, \\( qzxwvtnp=0, hjgrksla=mvdprqoe \\)."
    },
    "kernel_variant": {
      "question": "In the Euclidean space $\\mathbb{R}^{3}$ consider the two parabolas  \n\\[\n\\mathcal{C}_{1}\\colon z^{2}=5x,\\quad y=0 ,\n\\qquad  \n\\mathcal{C}_{2}\\colon y^{2}=7x,\\quad z=0 .\n\\]\n\nLet $\\Pi$ be the plane $y+z=0$.  \nA line $\\ell$ is called \\emph{admissible} if  \n\n(a) $\\ell$ meets $\\mathcal{C}_{1}$ and $\\mathcal{C}_{2}$ in two distinct points,  \n\n(b) $\\ell$ is parallel to $\\Pi$, that is, its direction vector $v$ satisfies $v\\!\\cdot\\!(0,1,1)=0$.  \n\nAnswer the following questions.\n\n(i)  Prove that admissible lines exist and determine an explicit one–parameter family  \n\\[\n\\mathfrak{L}\\;=\\;\\{\\ell(\\lambda)\\mid \\lambda\\in\\mathbb{R}\\setminus\\{0\\}\\}\n\\]\ncontaining \\emph{all} admissible lines.\n\n(ii)  Write every $\\ell(\\lambda)$ in homogeneous Plücker coordinates  \n\\[\n(p_{01}:p_{02}:p_{03}:p_{12}:p_{13}:p_{23})\\in\\mathbb{P}^{5},\n\\]\nshow that they satisfy both the Klein–quadric equation  \n\\[\np_{01}p_{23}+p_{02}p_{31}+p_{03}p_{12}=0 \\tag{$\\dagger$}\n\\]\nand the three independent linear relations  \n\\begin{align}\np_{02}+p_{03}&=0, \\tag{$\\ddagger_{1}$}\\\\\n5p_{12}+7p_{13}&=0, \\tag{$\\ddagger_{2}$}\\\\\n35p_{01}-2p_{23}&=0. \\tag{$\\ddagger_{3}$}\n\\end{align}\nLet  \n\\[\n\\Lambda\\;:=\\;\\bigl\\{(p)\\in\\mathbb{P}^{5}\\bigm|\\text{$(\\ddagger_{1})$–$(\\ddagger_{3})$ hold}\\bigr\\}\\cong\\mathbb{P}^{2},\n\\]\nand let $Z$ be the intersection of $\\Lambda$ with the Klein quadric, i.e. the common zero locus of $(\\dagger)$ and $(\\ddagger_{1})$–$(\\ddagger_{3})$.\n\n(1)  Prove that $Z$ is the \\emph{disjoint} union of the following three subsets  \n\n$\\bullet$ the Zariski–open part  \n\\[\nC\\;:=\\;\\Bigl\\{(p)\\in Z\\ \\Bigm|\\ p_{23}\\neq0\\Bigr\\},\n\\]\nwhich is a smooth conic $C\\cong\\mathbb{P}^{1}$ (whose points are exactly the lines of $\\mathfrak{L}$),  \n\n$\\bullet$ the Plücker point $P_{L}=(0:1:-1:0:0:0)$ representing the generator  \n\\[\nL\\colon x=0,\\; y+z=0,\n\\]\n\n$\\bullet$ the Plücker point $P_{\\infty}=(0:0:0:1:-5/7:0)$ representing an ideal line contained in the plane at infinity.  \n\n(2)  Show that neither $P_{L}$ nor $P_{\\infty}$ corresponds to an admissible line.\n\n(iii)  Eliminate the Plücker coordinates and show that the union of all admissible lines is  \n\\[\nS\\setminus L\\quad\\text{with}\\quad\nS\\colon x=\\dfrac{(y+z)(5y+7z)}{35}, \\tag{$\\star$}\n\\]\ni.e. the quadric surface $S$ minus the generator  \n\\[\nL\\colon x=0,\\;y+z=0\\;=\\;S\\cap\\Pi .\n\\]\nShow that every affine point of $S$ that does \\emph{not} lie on $L$ is contained in a unique admissible line, while $L$ is reached by no admissible line.\n\n(iv)  Verify that $S$ is a hyperbolic paraboloid.  Determine explicitly the\nsecond ruling  \n\\[\n\\widehat{\\mathfrak{L}}\\;=\\;\\{\\widehat{\\ell}(k)\\mid k\\in\\mathbb{R}\\},\n\\]\nand prove that no $\\widehat{\\ell}(k)$ is parallel to $\\Pi$ (hence no $\\widehat{\\ell}(k)$ is admissible), although every $\\widehat{\\ell}(k)$ meets $\\mathcal{C}_{1}$ and $\\mathcal{C}_{2}$.  (For $k=0$ the two intersections collapse to $(0,0,0)$, so $\\widehat{\\ell}(0)$ is not admissible.)\n\n(v)  Regard $S$ as the regular surface $(y,z)\\mapsto\\bigl(f(y,z),y,z\\bigr)$ with  \n\\[\nf(y,z)=\\dfrac{(y+z)(5y+7z)}{35}.\n\\]\n\n(1)  Compute the first and second fundamental forms and show that the\nGaussian curvature is  \n\\[\nK(y,z)\\;=\\;-\\dfrac{4}{35^{2}}\\bigl(1+\\|\\nabla f(y,z)\\|^{2}\\bigr)^{-2}\\;<\\;0\n\\quad\\forall\\,(y,z)\\in\\mathbb{R}^{2}.\n\\]\n\n(2)  Conclude that $S$ is a nowhere developable ruled surface:\nneither of its two rulings consists of developable directions.\n\n(3)  Find the mean curvature $H(y,z)$ and prove that it vanishes\nprecisely for those $(y,z)$ that satisfy  \n\\[\n5y^{2}+12yz+7z^{2}=3675. \\tag{$\\heartsuit$}\n\\]\n\nConsequently the set of minimal points of $S$ is the space curve  \n\\[\n\\Xi:=\\bigl\\{\\bigl(f(y,z),y,z\\bigr)\\bigm| (y,z)\\text{ satisfy }(\\heartsuit)\\bigr\\};\n\\]\nin particular, the origin $(0,0,0)$ is not a minimal point of $S$.\n\n\\bigskip",
      "solution": "\\textbf{Notation.}  Affinely embed $\\mathbb{R}^{3}$ in projective space $\\mathbb{P}^{3}$ with homogeneous coordinates $(w:x:y:z)$; $w\\neq0$ corresponds to the usual point $(x,y,z)$.\n\n\\medskip\n\\underline{\\textbf{Part (i) – Parametrisation of all admissible lines}}\n\nLet $P$ be the intersection of an admissible line with $\\mathcal{C}_{1}$ and $Q$ the intersection with $\\mathcal{C}_{2}$.  Write\n\\[\nP=(x_{1},0,z_{1}), \\qquad z_{1}^{2}=5x_{1},\\qquad \nQ=(x_{2},y_{2},0), \\qquad y_{2}^{2}=7x_{2}.\n\\]\nBecause the line is parallel to $\\Pi$ we must have $z_{1}=y_{2}=:\\lambda$ with $\\lambda\\neq0$.  Consequently\n\\[\nP=\\Bigl(\\dfrac{\\lambda^{2}}{5},0,\\lambda\\Bigr),\\qquad \nQ=\\Bigl(\\dfrac{\\lambda^{2}}{7},\\lambda,0\\Bigr). \\tag{1}\n\\]\nThe direction vector is\n\\[\nv=Q-P=\\Bigl(\\dfrac{\\lambda^{2}}{7}-\\dfrac{\\lambda^{2}}{5},\\lambda,-\\lambda\\Bigr)\n=\\Bigl(-\\dfrac{2\\lambda^{2}}{35},\\;\\lambda,\\;-\\lambda\\Bigr), \\tag{2}\n\\]\nand indeed $v\\!\\cdot\\!(0,1,1)=0$.  Conversely, condition (b) forces $z_{1}=y_{2}$, so every admissible line arises from some $\\lambda\\neq0$.  Taking an affine parameter $u$ along the line yields\n\\begin{align}\n\\ell(\\lambda)\\colon\\;(x,y,z)\n&=\\Bigl(\\dfrac{\\lambda^{2}}{5},0,\\lambda\\Bigr)\n  +u\\Bigl(-\\dfrac{2\\lambda^{2}}{35},\\lambda,-\\lambda\\Bigr)\\notag\\\\\n&=\\Bigl(\\dfrac{\\lambda^{2}(7-2u)}{35},\\;\\lambda u,\\;\\lambda(1-u)\\Bigr).\n\\tag{3}\n\\end{align}\nThus\n\\[\n\\mathfrak{L}\\;=\\;\\{\\ell(\\lambda)\\mid\\lambda\\in\\mathbb{R}\\setminus\\{0\\}\\}\n\\]\nis exactly the set of admissible lines.\n\n\\bigskip\n\\underline{\\textbf{Part (ii) – Plücker coordinates and the decomposition of $Z$}}\n\nWith homogeneous points $\\bar{P}=(1:\\lambda^{2}/5:0:\\lambda)$ and $\\bar{Q}=(1:\\lambda^{2}/7:\\lambda:0)$ the Plücker coordinates\n$p_{ij}=\\bar{P}_{i}\\bar{Q}_{j}-\\bar{P}_{j}\\bar{Q}_{i}$ are\n\\[\n\\begin{aligned}\np_{01}&=-\\dfrac{2\\lambda^{2}}{35}, & p_{02}&=\\lambda, & p_{03}&=-\\lambda,\\\\\np_{12}&=\\dfrac{\\lambda^{3}}{5},    & p_{13}&=-\\dfrac{\\lambda^{3}}{7}, & p_{23}&=-\\lambda^{2}. \\tag{4}\n\\end{aligned}\n\\]\nThey satisfy $(\\dagger)$ and $(\\ddagger_{1})$–$(\\ddagger_{3})$ by direct substitution.\n\n\\smallskip\nInside $\\mathbb{P}^{5}$ the three independent linear equations cut out the projective plane  \n\\[\n\\Lambda\\cong\\mathbb{P}^{2}.\n\\]\nIntroduce homogeneous coordinates on $\\Lambda$ by\n\\[\n(u:v:w):=(p_{02}:p_{12}:p_{01}). \\tag{5}\n\\]\nUsing $(\\ddagger_{1})$–$(\\ddagger_{3})$ we obtain\n\\[\np_{03}=-u,\\qquad p_{13}=-\\dfrac{5v}{7},\\qquad p_{23}= \\dfrac{35w}{2}. \\tag{6}\n\\]\nSubstituting these expressions into $(\\dagger)$ gives the quadratic\n\\[\n\\dfrac{35}{2}w^{2}-\\dfrac{2}{7}uv=0, \\qquad\\text{i.e.}\\qquad 245w^{2}-4uv=0. \\tag{7}\n\\]\nHence\n\\[\nZ=\\bigl\\{(u:v:w)\\in\\mathbb{P}^{2}\\bigm|245w^{2}-4uv=0\\bigr\\}. \\tag{8}\n\\]\n\n\\textbf{Decomposition of $Z$.}  \nDefine\n\\[\nC:=\\bigl\\{(u:v:w)\\in Z\\mid w\\neq0\\bigr\\}. \\tag{9}\n\\]\nBecause $w\\neq0$ one can de-homogenise the quadratic equation to obtain an affine equation of a smooth conic; hence $C$ is smooth and isomorphic to $\\mathbb{P}^{1}$.\n\nThe two points with $w=0$ are\n\\[\n(1:0:0)\\quad\\text{and}\\quad(0:1:0),\n\\]\ncorresponding (via (6)) to the Plücker points  \n\\[\nP_{L}=(0:1:-1:0:0:0),\\qquad\nP_{\\infty}=(0:0:0:1:-5/7:0).\n\\]\nThus\n\\[\nZ=C\\;\\dot{\\cup}\\;\\{P_{L}\\}\\;\\dot{\\cup}\\;\\{P_{\\infty}\\}. \\tag{10}\n\\]\nEmploying $w=-2\\lambda^{2}/35\\neq0$ we obtain\n\\[\nu=\\lambda,\\quad v=\\dfrac{\\lambda^{3}}{5},\\quad w=-\\dfrac{2\\lambda^{2}}{35},\n\\]\nwhich satisfies (7) and reproduces (4); consequently $C$ is exactly the\nPlücker image of $\\mathfrak{L}$ and no admissible line is represented by\n$P_{L}$ or $P_{\\infty}$.\n\n\\bigskip\n\\underline{\\textbf{Part (iii) – The surface swept out by $\\mathfrak{L}$}}\n\nFrom (3) we have the relations\n\\[\ny+z=\\lambda,\\qquad 5y+7z=\\lambda(7-2u). \\tag{11}\n\\]\nHence\n\\[\nx=\\dfrac{\\lambda^{2}(7-2u)}{35}\n     =\\dfrac{(y+z)(5y+7z)}{35}, \\tag{12}\n\\]\nso every admissible line lies on\n\\[\nS\\colon x=\\dfrac{(y+z)(5y+7z)}{35}. \\tag{13}\n\\]\nConversely, let $(x,y,z)\\in S$ with $(y,z)\\neq(0,0)$.  Put\n\\[\n\\lambda:=y+z,\\qquad u:=\\dfrac{y}{\\lambda}.\n\\]\nThen $(x,y,z)$ coincides with (3).  Because $\\lambda\\neq0$, the point\nlies on a \\emph{unique} admissible line.  \nPoints of $L\\colon x=0,\\;y+z=0$ satisfy $\\lambda=0$ and are not reached.  \nTherefore\n\\[\n\\bigcup_{\\ell\\in\\mathfrak{L}}\\ell \\;=\\;S\\setminus L. \\tag{14}\n\\]\n\n\\bigskip\n\\underline{\\textbf{Part (iv) – Hyperbolic paraboloid and the second ruling}}\n\nRewrite $(\\star)$ as\n\\[\n35x=(y+z)(5y+7z)=5y^{2}+12yz+7z^{2}. \\tag{15}\n\\]\nThe quadratic form on the right has determinant $5\\cdot7-6^{2}=-1<0$, hence $S$ is a hyperbolic paraboloid and therefore doubly ruled.\n\n\\smallskip\nFix $k\\in\\mathbb{R}$ and set\n\\[\ny(t)=7t,\\qquad z(t)=-5t+\\dfrac{k}{7}. \\tag{16}\n\\]\nThen $5y(t)+7z(t)=k$ and $y(t)+z(t)=2t+k/7$, giving\n\\[\nx(t)=\\dfrac{(2t+k/7)k}{35}=\\dfrac{2k}{35}t+\\dfrac{k^{2}}{245}. \\tag{17}\n\\]\nThus\n\\[\n\\widehat{\\ell}(k)\\colon\n(x,y,z)=\\Bigl(\\dfrac{2k}{35}t+\\dfrac{k^{2}}{245},\\;7t,\\;-5t+\\dfrac{k}{7}\\Bigr),\n\\quad t\\in\\mathbb{R}. \\tag{18}\n\\]\nThe direction vector is $\\widehat{v}=(2k/35,\\,7,\\,-5)$, and\n$\\widehat{v}\\!\\cdot\\!(0,1,1)=2\\neq0$, so no $\\widehat{\\ell}(k)$ is parallel to $\\Pi$.  If $k\\neq0$, (18) meets $\\mathcal{C}_{1}$ at $t=0$ and $\\mathcal{C}_{2}$ at $t=k/35$; for $k=0$ these merge to $(0,0,0)$.  Hence $\\widehat{\\mathfrak{L}}$ is the second ruling of $S$ and contains no admissible line.\n\n\\bigskip\n\\underline{\\textbf{Part (v) – Differential–geometric properties}}\n\nPut\n\\[\nr(y,z)=\\bigl(f(y,z),y,z\\bigr),\\qquad\nf(y,z)=\\dfrac{(y+z)(5y+7z)}{35}. \\tag{19}\n\\]\n\\textbf{First derivatives.}  \n\\[\nf_{y}=\\dfrac{10y+12z}{35},\\qquad \nf_{z}=\\dfrac{12y+14z}{35}.\n\\]\nSet $a:=f_{y}$, $b:=f_{z}$ and $W^{2}:=1+a^{2}+b^{2}$.\n\nFirst fundamental form  \n\\[\nE=1+a^{2},\\qquad F=ab,\\qquad G=1+b^{2}. \\tag{20}\n\\]\n\n\\textbf{Second derivatives.}  \n\\[\nf_{yy}=\\dfrac{2}{7},\\qquad f_{yz}=\\dfrac{12}{35},\\qquad f_{zz}=\\dfrac{2}{5}. \\tag{21}\n\\]\n\nSecond fundamental coefficients (with unit normal $N=(1,-a,-b)/W$):\n\\[\ne=\\dfrac{f_{yy}}{W},\\qquad \nf=\\dfrac{f_{yz}}{W},\\qquad \ng=\\dfrac{f_{zz}}{W}. \\tag{22}\n\\]\n\n\\textbf{Gaussian curvature.}  \n\\[\neg-f^{2}\n=\\dfrac{f_{yy}f_{zz}-f_{yz}^{2}}{W^{2}}\n=-\\dfrac{4}{35^{2}}\\dfrac{1}{W^{2}},\\qquad\nEG-F^{2}=W^{2}.\n\\]\nHence\n\\[\nK=\\dfrac{eg-f^{2}}{EG-F^{2}}\n=-\\dfrac{4}{35^{2}}\\dfrac{1}{W^{4}}\n=-\\dfrac{4}{35^{2}}\\bigl(1+\\|\\nabla f\\|^{2}\\bigr)^{-2}<0\\quad\\forall(y,z). \\tag{23}\n\\]\n\n\\textbf{Non–developability.}  \nFor a ruled surface, developability along a ruling would force $K\\equiv0$ on that ruling, impossible by (23).  Thus neither ruling is developable and $S$ is nowhere developable.\n\n\\textbf{Mean curvature.}  For a graph $(x,y,z)=(f(u,v),u,v)$\n\\[\nH=\\dfrac{(1+b^{2})f_{yy}-2ab\\,f_{yz}+(1+a^{2})f_{zz}}\n         {2\\bigl(1+a^{2}+b^{2}\\bigr)^{3/2}}. \\tag{24}\n\\]\nSubstituting the derivatives gives\n\\[\nH(y,z)=\\dfrac{12+7a^{2}+5b^{2}-12ab}\n              {35\\bigl(1+a^{2}+b^{2}\\bigr)^{3/2}}. \\tag{25}\n\\]\nClearing the denominator $35^{2}$ one finds\n\\[\nH=0\\;\\Longleftrightarrow\\;\n5y^{2}+12yz+7z^{2}=3675. \\tag{26}\n\\]\nBecause the quadratic form has determinant $-1$, (26) describes a real hyperbola in the $(y,z)$–plane.  Its image under $r$ is the space curve\n\\[\n\\Xi=\\bigl\\{\\bigl(f(y,z),y,z\\bigr)\\mid5y^{2}+12yz+7z^{2}=3675\\bigr\\}. \\tag{27}\n\\]\nThus the mean curvature vanishes exactly along $\\Xi$; in particular $H(0,0)=12/35>0$, so $(0,0,0)$ is not a minimal point.\n\n\\hfill$\\square$\n\n\\bigskip",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.383066",
        "was_fixed": false,
        "difficulty_analysis": "------------------------------------------------------------\n1.  Higher-level language.  The solution uses *projective geometry* and *Plücker coordinates*, tools absent from the original problem.\n\n2.  Extra conditions.  Besides eliminating two parameters as in the kernel problem, the solver must cope with **three** algebraic constraints—the Klein quadric and the two linear relations (‡)—and interpret their incidence-geometric meaning.\n\n3.  Multiple interacting concepts.  Classical analytic geometry (parametrising lines, eliminating parameters), projective/Grassmannian geometry (lines as points of the Klein quadric), algebraic-geometric reasoning (a conic as intersection of quadrics), and differential geometry (Gaussian curvature) all interact.\n\n4.  Deeper theory required.  Identifying the geometry of S as a *hyperbolic paraboloid*, finding the *second ruling*, and computing its *Gaussian curvature* demand results well beyond the undergraduate analytic manipulations sufficient for the original exercise.\n\n5.  More computations.  The Plücker-coordinate verification, elimination, curvature calculation, and ruling analysis pile several layers of non-trivial algebra onto the original two-parameter elimination.\n\nConsequently the enhanced variant is substantially harder and cannot be solved by a single-line pattern match; it imposes sophisticated techniques from several advanced areas of geometry."
      }
    },
    "original_kernel_variant": {
      "question": "In the Euclidean space $\\mathbb{R}^{3}$ consider the two parabolas  \n\\[\n\\mathcal{C}_{1}\\colon z^{2}=5x,\\quad y=0 ,\n\\qquad  \n\\mathcal{C}_{2}\\colon y^{2}=7x,\\quad z=0 .\n\\]\n\nLet $\\Pi$ be the plane $y+z=0$.  \nA line $\\ell$ is called \\emph{admissible} if  \n\n(a) $\\ell$ meets $\\mathcal{C}_{1}$ and $\\mathcal{C}_{2}$ in two distinct points,  \n\n(b) $\\ell$ is parallel to $\\Pi$, that is, its direction vector $v$ satisfies $v\\!\\cdot\\!(0,1,1)=0$.  \n\nAnswer the following questions.\n\n(i)  Prove that admissible lines exist and determine an explicit one–parameter family  \n\\[\n\\mathfrak{L}\\;=\\;\\{\\ell(\\lambda)\\mid \\lambda\\in\\mathbb{R}\\setminus\\{0\\}\\}\n\\]\ncontaining \\emph{all} admissible lines.\n\n(ii)  Write every $\\ell(\\lambda)$ in homogeneous Plücker coordinates  \n\\[\n(p_{01}:p_{02}:p_{03}:p_{12}:p_{13}:p_{23})\\in\\mathbb{P}^{5},\n\\]\nshow that they satisfy both the Klein–quadric equation  \n\\[\np_{01}p_{23}+p_{02}p_{31}+p_{03}p_{12}=0 \\tag{$\\dagger$}\n\\]\nand the three independent linear relations  \n\\begin{align}\np_{02}+p_{03}&=0, \\tag{$\\ddagger_{1}$}\\\\\n5p_{12}+7p_{13}&=0, \\tag{$\\ddagger_{2}$}\\\\\n35p_{01}-2p_{23}&=0. \\tag{$\\ddagger_{3}$}\n\\end{align}\nLet  \n\\[\n\\Lambda\\;:=\\;\\bigl\\{(p)\\in\\mathbb{P}^{5}\\bigm|\\text{$(\\ddagger_{1})$–$(\\ddagger_{3})$ hold}\\bigr\\}\\cong\\mathbb{P}^{2},\n\\]\nand let $Z$ be the intersection of $\\Lambda$ with the Klein quadric, i.e. the common zero locus of $(\\dagger)$ and $(\\ddagger_{1})$–$(\\ddagger_{3})$.\n\n(1)  Prove that $Z$ is the \\emph{disjoint} union of the following three subsets  \n\n$\\bullet$ the Zariski–open part  \n\\[\nC\\;:=\\;\\Bigl\\{(p)\\in Z\\ \\Bigm|\\ p_{23}\\neq0\\Bigr\\},\n\\]\nwhich is a smooth conic $C\\cong\\mathbb{P}^{1}$ (whose points are exactly the lines of $\\mathfrak{L}$),  \n\n$\\bullet$ the Plücker point $P_{L}=(0:1:-1:0:0:0)$ representing the generator  \n\\[\nL\\colon x=0,\\; y+z=0,\n\\]\n\n$\\bullet$ the Plücker point $P_{\\infty}=(0:0:0:1:-5/7:0)$ representing an ideal line contained in the plane at infinity.  \n\n(2)  Show that neither $P_{L}$ nor $P_{\\infty}$ corresponds to an admissible line.\n\n(iii)  Eliminate the Plücker coordinates and show that the union of all admissible lines is  \n\\[\nS\\setminus L\\quad\\text{with}\\quad\nS\\colon x=\\dfrac{(y+z)(5y+7z)}{35}, \\tag{$\\star$}\n\\]\ni.e. the quadric surface $S$ minus the generator  \n\\[\nL\\colon x=0,\\;y+z=0\\;=\\;S\\cap\\Pi .\n\\]\nShow that every affine point of $S$ that does \\emph{not} lie on $L$ is contained in a unique admissible line, while $L$ is reached by no admissible line.\n\n(iv)  Verify that $S$ is a hyperbolic paraboloid.  Determine explicitly the\nsecond ruling  \n\\[\n\\widehat{\\mathfrak{L}}\\;=\\;\\{\\widehat{\\ell}(k)\\mid k\\in\\mathbb{R}\\},\n\\]\nand prove that no $\\widehat{\\ell}(k)$ is parallel to $\\Pi$ (hence no $\\widehat{\\ell}(k)$ is admissible), although every $\\widehat{\\ell}(k)$ meets $\\mathcal{C}_{1}$ and $\\mathcal{C}_{2}$.  (For $k=0$ the two intersections collapse to $(0,0,0)$, so $\\widehat{\\ell}(0)$ is not admissible.)\n\n(v)  Regard $S$ as the regular surface $(y,z)\\mapsto\\bigl(f(y,z),y,z\\bigr)$ with  \n\\[\nf(y,z)=\\dfrac{(y+z)(5y+7z)}{35}.\n\\]\n\n(1)  Compute the first and second fundamental forms and show that the\nGaussian curvature is  \n\\[\nK(y,z)\\;=\\;-\\dfrac{4}{35^{2}}\\bigl(1+\\|\\nabla f(y,z)\\|^{2}\\bigr)^{-2}\\;<\\;0\n\\quad\\forall\\,(y,z)\\in\\mathbb{R}^{2}.\n\\]\n\n(2)  Conclude that $S$ is a nowhere developable ruled surface:\nneither of its two rulings consists of developable directions.\n\n(3)  Find the mean curvature $H(y,z)$ and prove that it vanishes\nprecisely for those $(y,z)$ that satisfy  \n\\[\n5y^{2}+12yz+7z^{2}=3675. \\tag{$\\heartsuit$}\n\\]\n\nConsequently the set of minimal points of $S$ is the space curve  \n\\[\n\\Xi:=\\bigl\\{\\bigl(f(y,z),y,z\\bigr)\\bigm| (y,z)\\text{ satisfy }(\\heartsuit)\\bigr\\};\n\\]\nin particular, the origin $(0,0,0)$ is not a minimal point of $S$.\n\n\\bigskip",
      "solution": "\\textbf{Notation.}  Affinely embed $\\mathbb{R}^{3}$ in projective space $\\mathbb{P}^{3}$ with homogeneous coordinates $(w:x:y:z)$; $w\\neq0$ corresponds to the usual point $(x,y,z)$.\n\n\\medskip\n\\underline{\\textbf{Part (i) – Parametrisation of all admissible lines}}\n\nLet $P$ be the intersection of an admissible line with $\\mathcal{C}_{1}$ and $Q$ the intersection with $\\mathcal{C}_{2}$.  Write\n\\[\nP=(x_{1},0,z_{1}), \\qquad z_{1}^{2}=5x_{1},\\qquad \nQ=(x_{2},y_{2},0), \\qquad y_{2}^{2}=7x_{2}.\n\\]\nBecause the line is parallel to $\\Pi$ we must have $z_{1}=y_{2}=:\\lambda$ with $\\lambda\\neq0$.  Consequently\n\\[\nP=\\Bigl(\\dfrac{\\lambda^{2}}{5},0,\\lambda\\Bigr),\\qquad \nQ=\\Bigl(\\dfrac{\\lambda^{2}}{7},\\lambda,0\\Bigr). \\tag{1}\n\\]\nThe direction vector is\n\\[\nv=Q-P=\\Bigl(\\dfrac{\\lambda^{2}}{7}-\\dfrac{\\lambda^{2}}{5},\\lambda,-\\lambda\\Bigr)\n=\\Bigl(-\\dfrac{2\\lambda^{2}}{35},\\;\\lambda,\\;-\\lambda\\Bigr), \\tag{2}\n\\]\nand indeed $v\\!\\cdot\\!(0,1,1)=0$.  Conversely, condition (b) forces $z_{1}=y_{2}$, so every admissible line arises from some $\\lambda\\neq0$.  Taking an affine parameter $u$ along the line yields\n\\begin{align}\n\\ell(\\lambda)\\colon\\;(x,y,z)\n&=\\Bigl(\\dfrac{\\lambda^{2}}{5},0,\\lambda\\Bigr)\n  +u\\Bigl(-\\dfrac{2\\lambda^{2}}{35},\\lambda,-\\lambda\\Bigr)\\notag\\\\\n&=\\Bigl(\\dfrac{\\lambda^{2}(7-2u)}{35},\\;\\lambda u,\\;\\lambda(1-u)\\Bigr).\n\\tag{3}\n\\end{align}\nThus\n\\[\n\\mathfrak{L}\\;=\\;\\{\\ell(\\lambda)\\mid\\lambda\\in\\mathbb{R}\\setminus\\{0\\}\\}\n\\]\nis exactly the set of admissible lines.\n\n\\bigskip\n\\underline{\\textbf{Part (ii) – Plücker coordinates and the decomposition of $Z$}}\n\nWith homogeneous points $\\bar{P}=(1:\\lambda^{2}/5:0:\\lambda)$ and $\\bar{Q}=(1:\\lambda^{2}/7:\\lambda:0)$ the Plücker coordinates\n$p_{ij}=\\bar{P}_{i}\\bar{Q}_{j}-\\bar{P}_{j}\\bar{Q}_{i}$ are\n\\[\n\\begin{aligned}\np_{01}&=-\\dfrac{2\\lambda^{2}}{35}, & p_{02}&=\\lambda, & p_{03}&=-\\lambda,\\\\\np_{12}&=\\dfrac{\\lambda^{3}}{5},    & p_{13}&=-\\dfrac{\\lambda^{3}}{7}, & p_{23}&=-\\lambda^{2}. \\tag{4}\n\\end{aligned}\n\\]\nThey satisfy $(\\dagger)$ and $(\\ddagger_{1})$–$(\\ddagger_{3})$ by direct substitution.\n\n\\smallskip\nInside $\\mathbb{P}^{5}$ the three independent linear equations cut out the projective plane  \n\\[\n\\Lambda\\cong\\mathbb{P}^{2}.\n\\]\nIntroduce homogeneous coordinates on $\\Lambda$ by\n\\[\n(u:v:w):=(p_{02}:p_{12}:p_{01}). \\tag{5}\n\\]\nUsing $(\\ddagger_{1})$–$(\\ddagger_{3})$ we obtain\n\\[\np_{03}=-u,\\qquad p_{13}=-\\dfrac{5v}{7},\\qquad p_{23}= \\dfrac{35w}{2}. \\tag{6}\n\\]\nSubstituting these expressions into $(\\dagger)$ gives the quadratic\n\\[\n\\dfrac{35}{2}w^{2}-\\dfrac{2}{7}uv=0, \\qquad\\text{i.e.}\\qquad 245w^{2}-4uv=0. \\tag{7}\n\\]\nHence\n\\[\nZ=\\bigl\\{(u:v:w)\\in\\mathbb{P}^{2}\\bigm|245w^{2}-4uv=0\\bigr\\}. \\tag{8}\n\\]\n\n\\textbf{Decomposition of $Z$.}  \nDefine\n\\[\nC:=\\bigl\\{(u:v:w)\\in Z\\mid w\\neq0\\bigr\\}. \\tag{9}\n\\]\nBecause $w\\neq0$ one can de-homogenise the quadratic equation to obtain an affine equation of a smooth conic; hence $C$ is smooth and isomorphic to $\\mathbb{P}^{1}$.\n\nThe two points with $w=0$ are\n\\[\n(1:0:0)\\quad\\text{and}\\quad(0:1:0),\n\\]\ncorresponding (via (6)) to the Plücker points  \n\\[\nP_{L}=(0:1:-1:0:0:0),\\qquad\nP_{\\infty}=(0:0:0:1:-5/7:0).\n\\]\nThus\n\\[\nZ=C\\;\\dot{\\cup}\\;\\{P_{L}\\}\\;\\dot{\\cup}\\;\\{P_{\\infty}\\}. \\tag{10}\n\\]\nEmploying $w=-2\\lambda^{2}/35\\neq0$ we obtain\n\\[\nu=\\lambda,\\quad v=\\dfrac{\\lambda^{3}}{5},\\quad w=-\\dfrac{2\\lambda^{2}}{35},\n\\]\nwhich satisfies (7) and reproduces (4); consequently $C$ is exactly the\nPlücker image of $\\mathfrak{L}$ and no admissible line is represented by\n$P_{L}$ or $P_{\\infty}$.\n\n\\bigskip\n\\underline{\\textbf{Part (iii) – The surface swept out by $\\mathfrak{L}$}}\n\nFrom (3) we have the relations\n\\[\ny+z=\\lambda,\\qquad 5y+7z=\\lambda(7-2u). \\tag{11}\n\\]\nHence\n\\[\nx=\\dfrac{\\lambda^{2}(7-2u)}{35}\n     =\\dfrac{(y+z)(5y+7z)}{35}, \\tag{12}\n\\]\nso every admissible line lies on\n\\[\nS\\colon x=\\dfrac{(y+z)(5y+7z)}{35}. \\tag{13}\n\\]\nConversely, let $(x,y,z)\\in S$ with $(y,z)\\neq(0,0)$.  Put\n\\[\n\\lambda:=y+z,\\qquad u:=\\dfrac{y}{\\lambda}.\n\\]\nThen $(x,y,z)$ coincides with (3).  Because $\\lambda\\neq0$, the point\nlies on a \\emph{unique} admissible line.  \nPoints of $L\\colon x=0,\\;y+z=0$ satisfy $\\lambda=0$ and are not reached.  \nTherefore\n\\[\n\\bigcup_{\\ell\\in\\mathfrak{L}}\\ell \\;=\\;S\\setminus L. \\tag{14}\n\\]\n\n\\bigskip\n\\underline{\\textbf{Part (iv) – Hyperbolic paraboloid and the second ruling}}\n\nRewrite $(\\star)$ as\n\\[\n35x=(y+z)(5y+7z)=5y^{2}+12yz+7z^{2}. \\tag{15}\n\\]\nThe quadratic form on the right has determinant $5\\cdot7-6^{2}=-1<0$, hence $S$ is a hyperbolic paraboloid and therefore doubly ruled.\n\n\\smallskip\nFix $k\\in\\mathbb{R}$ and set\n\\[\ny(t)=7t,\\qquad z(t)=-5t+\\dfrac{k}{7}. \\tag{16}\n\\]\nThen $5y(t)+7z(t)=k$ and $y(t)+z(t)=2t+k/7$, giving\n\\[\nx(t)=\\dfrac{(2t+k/7)k}{35}=\\dfrac{2k}{35}t+\\dfrac{k^{2}}{245}. \\tag{17}\n\\]\nThus\n\\[\n\\widehat{\\ell}(k)\\colon\n(x,y,z)=\\Bigl(\\dfrac{2k}{35}t+\\dfrac{k^{2}}{245},\\;7t,\\;-5t+\\dfrac{k}{7}\\Bigr),\n\\quad t\\in\\mathbb{R}. \\tag{18}\n\\]\nThe direction vector is $\\widehat{v}=(2k/35,\\,7,\\,-5)$, and\n$\\widehat{v}\\!\\cdot\\!(0,1,1)=2\\neq0$, so no $\\widehat{\\ell}(k)$ is parallel to $\\Pi$.  If $k\\neq0$, (18) meets $\\mathcal{C}_{1}$ at $t=0$ and $\\mathcal{C}_{2}$ at $t=k/35$; for $k=0$ these merge to $(0,0,0)$.  Hence $\\widehat{\\mathfrak{L}}$ is the second ruling of $S$ and contains no admissible line.\n\n\\bigskip\n\\underline{\\textbf{Part (v) – Differential–geometric properties}}\n\nPut\n\\[\nr(y,z)=\\bigl(f(y,z),y,z\\bigr),\\qquad\nf(y,z)=\\dfrac{(y+z)(5y+7z)}{35}. \\tag{19}\n\\]\n\\textbf{First derivatives.}  \n\\[\nf_{y}=\\dfrac{10y+12z}{35},\\qquad \nf_{z}=\\dfrac{12y+14z}{35}.\n\\]\nSet $a:=f_{y}$, $b:=f_{z}$ and $W^{2}:=1+a^{2}+b^{2}$.\n\nFirst fundamental form  \n\\[\nE=1+a^{2},\\qquad F=ab,\\qquad G=1+b^{2}. \\tag{20}\n\\]\n\n\\textbf{Second derivatives.}  \n\\[\nf_{yy}=\\dfrac{2}{7},\\qquad f_{yz}=\\dfrac{12}{35},\\qquad f_{zz}=\\dfrac{2}{5}. \\tag{21}\n\\]\n\nSecond fundamental coefficients (with unit normal $N=(1,-a,-b)/W$):\n\\[\ne=\\dfrac{f_{yy}}{W},\\qquad \nf=\\dfrac{f_{yz}}{W},\\qquad \ng=\\dfrac{f_{zz}}{W}. \\tag{22}\n\\]\n\n\\textbf{Gaussian curvature.}  \n\\[\neg-f^{2}\n=\\dfrac{f_{yy}f_{zz}-f_{yz}^{2}}{W^{2}}\n=-\\dfrac{4}{35^{2}}\\dfrac{1}{W^{2}},\\qquad\nEG-F^{2}=W^{2}.\n\\]\nHence\n\\[\nK=\\dfrac{eg-f^{2}}{EG-F^{2}}\n=-\\dfrac{4}{35^{2}}\\dfrac{1}{W^{4}}\n=-\\dfrac{4}{35^{2}}\\bigl(1+\\|\\nabla f\\|^{2}\\bigr)^{-2}<0\\quad\\forall(y,z). \\tag{23}\n\\]\n\n\\textbf{Non–developability.}  \nFor a ruled surface, developability along a ruling would force $K\\equiv0$ on that ruling, impossible by (23).  Thus neither ruling is developable and $S$ is nowhere developable.\n\n\\textbf{Mean curvature.}  For a graph $(x,y,z)=(f(u,v),u,v)$\n\\[\nH=\\dfrac{(1+b^{2})f_{yy}-2ab\\,f_{yz}+(1+a^{2})f_{zz}}\n         {2\\bigl(1+a^{2}+b^{2}\\bigr)^{3/2}}. \\tag{24}\n\\]\nSubstituting the derivatives gives\n\\[\nH(y,z)=\\dfrac{12+7a^{2}+5b^{2}-12ab}\n              {35\\bigl(1+a^{2}+b^{2}\\bigr)^{3/2}}. \\tag{25}\n\\]\nClearing the denominator $35^{2}$ one finds\n\\[\nH=0\\;\\Longleftrightarrow\\;\n5y^{2}+12yz+7z^{2}=3675. \\tag{26}\n\\]\nBecause the quadratic form has determinant $-1$, (26) describes a real hyperbola in the $(y,z)$–plane.  Its image under $r$ is the space curve\n\\[\n\\Xi=\\bigl\\{\\bigl(f(y,z),y,z\\bigr)\\mid5y^{2}+12yz+7z^{2}=3675\\bigr\\}. \\tag{27}\n\\]\nThus the mean curvature vanishes exactly along $\\Xi$; in particular $H(0,0)=12/35>0$, so $(0,0,0)$ is not a minimal point.\n\n\\hfill$\\square$\n\n\\bigskip",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.330653",
        "was_fixed": false,
        "difficulty_analysis": "------------------------------------------------------------\n1.  Higher-level language.  The solution uses *projective geometry* and *Plücker coordinates*, tools absent from the original problem.\n\n2.  Extra conditions.  Besides eliminating two parameters as in the kernel problem, the solver must cope with **three** algebraic constraints—the Klein quadric and the two linear relations (‡)—and interpret their incidence-geometric meaning.\n\n3.  Multiple interacting concepts.  Classical analytic geometry (parametrising lines, eliminating parameters), projective/Grassmannian geometry (lines as points of the Klein quadric), algebraic-geometric reasoning (a conic as intersection of quadrics), and differential geometry (Gaussian curvature) all interact.\n\n4.  Deeper theory required.  Identifying the geometry of S as a *hyperbolic paraboloid*, finding the *second ruling*, and computing its *Gaussian curvature* demand results well beyond the undergraduate analytic manipulations sufficient for the original exercise.\n\n5.  More computations.  The Plücker-coordinate verification, elimination, curvature calculation, and ruling analysis pile several layers of non-trivial algebra onto the original two-parameter elimination.\n\nConsequently the enhanced variant is substantially harder and cannot be solved by a single-line pattern match; it imposes sophisticated techniques from several advanced areas of geometry."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}