Okay, let's see. I need to find the value of c for which the given circle equation has a radius of 1. Hmm, the equation is x² -10x + y² +6y + c = 0. I remember that the general form of a circle equation is x² + y² + Dx + Ey + F = 0, and that it can be rewritten in standard form by completing the square. The standard form is (x - h)² + (y - k)² = r², where (h,k) is the center and r is the radius. So, I need to complete the square for both x and y terms in the given equation to get it into standard form, and then set the radius equal to 1 to solve for c.

First, let's group the x terms and y terms together. The equation is x² -10x + y² +6y + c = 0. Let me rearrange it as (x² -10x) + (y² +6y) + c = 0. Now, to complete the square for the x terms: x² -10x. The coefficient of x is -10. To complete the square, I take half of -10, which is -5, and square it, getting 25. So, x² -10x can be written as (x -5)² -25. Similarly, for the y terms: y² +6y. The coefficient of y is 6. Half of 6 is 3, and squaring that gives 9. So, y² +6y becomes (y +3)² -9. 

Substituting these back into the equation: (x -5)² -25 + (y +3)² -9 + c = 0. Combine the constants: -25 -9 + c = -34 + c. So, the equation becomes (x -5)² + (y +3)² + (c -34) = 0. To get it into standard form, move the constant term to the other side: (x -5)² + (y +3)² = 34 - c. 

Since the standard form is (x - h)² + (y - k)² = r², the radius squared is 34 - c. But the problem states that the radius is 1, so r² = 1² = 1. Therefore, 34 - c = 1. Solving for c: c = 34 -1 = 33. 

Wait, let me check my steps again to make sure I didn't make a mistake. Starting with the original equation: x² -10x + y² +6y + c = 0. Completed the square for x: x² -10x = (x -5)² -25. For y: y² +6y = (y +3)² -9. Substituting back gives (x -5)² -25 + (y +3)² -9 + c = 0. Combining constants: -25 -9 + c = -34 + c. So, (x -5)² + (y +3)² = 34 - c. Setting radius 1: 34 - c = 1 → c = 33. 

Yes, that seems right. Let me verify by plugging c =33 back into the original equation. So, x² -10x + y² +6y +33 =0. Completing the square again: (x² -10x +25) + (y² +6y +9) = -33 +25 +9. Which simplifies to (x -5)² + (y +3)² = (25 +9) -33 = 34 -33 =1. So, radius squared is 1, radius is 1. That checks out. 

Therefore, the value of c is 33.
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