Okay, so I need to find the equation of the set of image points after reflecting each point on the circle x² + y² = 25 over the point (4,1). Then, express this equation in the form x² + a y² + b x + c y + d = 0 and find the ordered quadruple (a, b, c, d). Let me break this down step by step.

First, reflection over a point. Wait, reflection over a point is different from reflection over a line. Reflecting a point (x, y) over another point (h, k) should result in a new point (2h - x, 2k - y). Let me confirm that. If you have a point A and you reflect it over point B, the reflection A' is such that B is the midpoint of segment AA'. So, if B is (h, k), then the coordinates of A' would be (2h - x, 2k - y). Yes, that makes sense. So, to get the image of any point (x, y) on the original circle, we replace x with 2*4 - x = 8 - x and y with 2*1 - y = 2 - y.

So, if the original circle is x² + y² = 25, after reflection, the image points (x', y') satisfy x' = 8 - x and y' = 2 - y. Therefore, we can express the original coordinates (x, y) in terms of the image coordinates (x', y'):

x = 8 - x'
y = 2 - y'

Now, substitute these into the original circle equation:

x² + y² = 25
(8 - x')² + (2 - y')² = 25

Let me expand this equation:

(8 - x')² = 64 - 16x' + x'²
(2 - y')² = 4 - 4y' + y'²

Adding them together:

64 - 16x' + x'² + 4 - 4y' + y'² = 25

Combine like terms:

x'² + y'² - 16x' - 4y' + 64 + 4 - 25 = 0

Calculate constants: 64 + 4 = 68; 68 - 25 = 43

So:

x'² + y'² - 16x' - 4y' + 43 = 0

But the problem states that the image equation is of the form x² + a y² + b x + c y + d = 0. Wait, in the problem, the equation is written as x² + a y² + b x + c y + d = 0. But in our case, after substitution, we have x'² + y'² -16x' -4y' +43 =0. Comparing this to the given form x² + a y² + b x + c y + d =0, we can see that a is 1 because the coefficient of y² is 1, b is -16, c is -4, and d is 43. So the ordered quadruple would be (1, -16, -4, 43). 

Wait, but let me double-check. Let me verify the reflection process again. If reflecting a point (x, y) over the point (4,1), the image point (x', y') should satisfy (4,1) being the midpoint between (x, y) and (x', y'). Therefore:

4 = (x + x')/2 => x + x' = 8 => x' = 8 - x

1 = (y + y')/2 => y + y' = 2 => y' = 2 - y

Yes, that's correct. So substituting x = 8 - x', y = 2 - y' into the original equation gives the image equation.

Let me recheck the expansion:

(8 - x')² = 64 -16x' +x'²

(2 - y')² = 4 -4y' + y'²

Adding these:

64 -16x' +x'² +4 -4y' +y'² =25

Combine constants: 64 +4=68, 68-25=43

So x'² + y'² -16x' -4y' +43=0

Which matches the form x² +a y² +b x +c y +d=0. Comparing term by term:

x² term: coefficient is 1 (same as x'², since the equation is in x' and y')

y² term: coefficient is 1 (since the original equation had y² with coefficient 1, and we have y'² as well)

x term: coefficient is -16 (from -16x')

y term: coefficient is -4 (from -4y')

Constant term: 43

Therefore, the ordered quadruple (a, b, c, d) is (1, -16, -4, 43). Hmm, but let me check the problem statement again. It says "the set of image points satisfies the equation x² + a y² + b x + c y + d =0." Wait, the original equation is x² + y² =25, which is in x and y. The image points are (x', y'), so substituting x=8 -x', y=2 - y' into x² + y²=25 gives (8 -x')² + (2 - y')²=25, which is x'² + y'² -16x' -4y' +43=0. So the image equation is x'² + y'² -16x' -4y' +43=0. So in terms of x' and y', the equation is x'² +1*y'² -16x' -4y' +43=0. Therefore, a=1, b=-16, c=-4, d=43. So (a, b, c, d) is (1, -16, -4, 43).

But let me check if there's any mistake here. For example, sometimes when you substitute coordinates, the signs can be tricky. Let me confirm again:

Original equation: x² + y² =25

After reflection: x' =8 -x, y'=2 -y

So x=8 -x', y=2 -y'

Substitute into original equation:

(8 -x')² + (2 -y')² =25

Expanding:

(64 -16x' +x'²) + (4 -4y' +y'²) =25

64 +4 =68; 68 -25=43

So x'² + y'² -16x' -4y' +43=0

Yes, that seems right. So in terms of x' and y', the coefficients are a=1, b=-16, c=-4, d=43. Therefore, the answer is (1, -16, -4, 43). 

Wait, but the problem says "the set of image points satisfies the equation x² + a y² + b x + c y + d =0." So in the equation, the x term is multiplied by 1, the y² term is multiplied by a, but in the image equation, x'² + y'² -16x' -4y' +43=0, which would correspond to x² + y² + (-16)x + (-4)y +43=0. Therefore, in the given form x² + a y² + b x + c y + d=0, the coefficients are:

- For x²: 1 (matches)
- For y²: 1 (so a=1)
- For x: -16 (so b=-16)
- For y: -4 (so c=-4)
- Constant term:43 (so d=43)

Therefore, the ordered quadruple is (1, -16, -4, 43). 

I think that's correct. Let me check once more with another approach. Suppose we take a point on the original circle, reflect it over (4,1), and see if it satisfies the derived equation.

Take a point on the original circle, say (5,0). Its reflection over (4,1) would be (8 -5, 2 -0) = (3, 2). Let's check if (3,2) satisfies the equation x² + y² -16x -4y +43=0.

Compute 3² +2² -16*3 -4*2 +43 =9 +4 -48 -8 +43= (13) -56 +43= (13 +43) -56=56 -56=0. Yes, it works.

Another test: original point (0,5). Its reflection is (8 -0, 2 -5)= (8, -3). Check equation: 8² + (-3)² -16*8 -4*(-3) +43=64 +9 -128 +12 +43= (73) -128 +55= (73 +55) -128=128 -128=0. Correct.

Another test: original point (3,4). It's on the circle because 3² +4²=25. Reflection over (4,1): (8 -3, 2 -4)=(5, -2). Check equation:5² + (-2)² -16*5 -4*(-2)+43=25 +4 -80 +8 +43= (29) -80 +51= (29 +51) -80=80 -80=0. Correct.

Therefore, the equation seems to be correct. So the ordered quadruple is (1, -16, -4, 43). 

I think that's the answer. Let me just check if the problem states "the set of image points satisfies the equation x² + a y² + b x + c y + d =0." So in the image points, which are (x', y'), the equation is x'² + y'² -16x' -4y' +43=0. So when written in terms of x and y (assuming x and y are the image coordinates), it's x² + y² -16x -4y +43=0. Therefore, a=1, b=-16, c=-4, d=43. So the quadruple is (1, -16, -4, 43). 

Yes, that seems to be correct. I don't see any mistakes in the reasoning or calculations. The tests with specific points worked, so the equation is correct. Therefore, the ordered quadruple is (1, -16, -4, 43).
Therefore, the ordered quadruple $(a, b, c, d)$ of real numbers is $\boxed{(1, -16, -4, 43)}$.
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