<Question>
If each point of the circle $x^2 + y^2 = 25$ is reflected in the point $(4,1),$ the set of image points satisfies the equation
\[x^2 + ay^2 + bx + cy + d = 0.\]Compute the ordered quadruple $(a,b,c,d)$ of real numbers.
</Question>

<think>
<Plan_1>
Okay, so I need to find the equation of the set of points obtained by 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). Hmm, reflection over a point. Let me recall what that means.

<prompt_1.1> Recall the concept of reflection over a point and determine the formula for the image of a point (x, y) when reflected over (h, k). </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Reflection of a point (x, y) over another point (h, k) results in a point (2h - x, 2k - y). Is that right? Because reflecting over a point is like taking the point as the midpoint between the original and the image. So if (h, k) is the midpoint, then the image (x', y') should satisfy h = (x + x')/2 and k = (y + y')/2. Solving for x' and y' gives x' = 2h - x and y' = 2k - y. Yeah, that seems correct.

So in this case, the reflection over (4,1) would transform each point (x, y) on the original circle to (2*4 - x, 2*1 - y) = (8 - x, 2 - y).
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Therefore, the image of the circle x² + y² = 25 under this reflection should be another circle, right? Because reflection is a linear transformation (well, affine actually, since it's a translation combined with reflection).
But let's verify that.

First, let me denote the original point as (X, Y) lying on the circle X² + Y² = 25.
Then the reflected point (x, y) is given by x = 8 - X and y = 2 - Y. So we can express X and Y in terms of x and y:

<prompt_2.1> Express the original coordinates (X, Y) in terms of the reflected coordinates (x, y) and substitute into the original circle equation to find the equation of the image. </prompt_2.1>
-----

Hmm, but let me check if this is correct. Let me verify with a sample point.

<prompt_2.2> Verify the solution by checking if the sample points (0,0), (5,0), and (0,5) lie on the transformed circle using the derived equation. </prompt_2.2>
-----

Alternatively, maybe there's another way to approach this. For example, the reflection of a circle over a point is equivalent to translating the center of the circle by twice the vector from the original center to the reflection point. Wait, reflection over a point is equivalent to a point reflection, which is inversion through a point. So if the original circle has center (0,0) and radius 5, then reflecting the center (0,0) over the point (4,1) would give the new center as (8,2). Because the reflection of the center is (2*4 - 0, 2*1 - 0) = (8,2). So the image circle should have center (8,2) and the same radius 5.

<prompt_2.3> Use the geometric approach to find the equation of the transformed circle and compare it with the previous result. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
X = 8 - x

Y = 2 - y

Since (X, Y) is on the original circle, substituting into X² + Y² = 25 gives:

(8 - x)² + (2 - y)² = 25

Expanding this equation should give the equation of the image circle.
</execution_2.1>

<execution_2.2>
Let's take the original point (0,0) on the circle x² + y² = 25. Reflecting (0,0) over (4,1) should give (8,2). So (8,2) should lie on the image circle. Let's check if (8,2) satisfies the equation - but wait, we need to check the transformed equation. Wait, actually, the image points are (8 - X, 2 - Y), so (8,2) is indeed the image of (0,0). Then the image circle should have center (8,2) and radius 5. So expanding (x - 8)² + (y - 2)² = 25. Let's expand that:

x² - 16x + 64 + y² -4y +4 =25

x² + y² -16x -4y +68 -25=0

x² + y² -16x -4y +43=0

Which matches the previous result. So expanding this equation, we get x² + y² -16x -4y +43 = 0. Comparing with the given form x² + a y² +b x +c y +d =0, since the coefficient of y² is 1, same as x², so a=1, b=-16, c=-4, d=43. Therefore, the ordered quadruple is (1, -16, -4, 43).
</execution_2.2>

<execution_2.3>
Therefore, the equation is (x - 8)² + (y - 2)² = 25. Expanding this:

x² - 16x + 64 + y² - 4y + 4 = 25

Which simplifies to x² + y² -16x -4y +68 = 25

Then x² + y² -16x -4y +43 = 0

Which is the same as before. So the equation is x² + y² -16x -4y +43 = 0. Therefore, comparing to the given form x² + a y² +b x +c y +d =0, since the coefficient of y² is 1, same as x², so a=1, b=-16, c=-4, d=43.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Expanding (8 - x)²:

<prompt_3.1> Expand and simplify the equation obtained from substituting the reflected coordinates into the original circle equation. </prompt_3.1>
-----

Based on execution_2.3:
So the answer is (1, -16, -4, 43). Therefore, the ordered quadruple is (1, -16, -4, 43).
I think this is correct.

<prompt_3.2> Finalize the solution and confirm the answer by ensuring all steps and checks are consistent. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
= 64 - 16x + x²

Similarly, (2 - y)² = 4 - 4y + y²

So adding them together:

(64 -16x + x²) + (4 -4y + y²) = 25

Combine like terms:

x² + y² -16x -4y + 68 = 25

Subtract 25 from both sides:

x² + y² -16x -4y + 43 = 0

Wait, but the problem states the general form x² + a y² + b x + c y + d = 0. In our case, the coefficients of x² and y² are both 1. So comparing, we have:

x² + 1*y² -16x -4y +43 = 0

Therefore, a = 1, b = -16, c = -4, d = 43. So the ordered quadruple is (1, -16, -4, 43).
</execution_3.1>

<execution_3.2>
Let me just check the expansion again to make sure I didn't make any arithmetic errors.

(8 - x)² = 64 - 16x + x²

(2 - y)² = 4 -4y + y²

Adding these together: x² + y² -16x -4y + 68 =25

Subtract 25: x² + y² -16x -4y +43=0. Yes, correct.

So the coefficients are a=1, b=-16, c=-4, d=43. Therefore, the ordered quadruple is (1, -16, -4, 43).
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
So the answer is (1, -16, -4, 43). Hmm, but let me check if there's a possibility that the problem expects a different form or if I made a mistake in the reflection formula.

<prompt_4.1> Verify the solution by checking the reflection formula and expansion of the equation. </prompt_4.1>
-----

Based on execution_3.2:
**Final Answer**
The ordered quadruple is \boxed{(1, -16, -4, 43)}.
</Plan_4>

<Execution_4>
<execution_4.1>
Wait, reflection over a point: if you reflect a point (x, y) over (h, k), the formula is (2h - x, 2k - y). So in this case, h=4, k=1, so (8 - x, 2 - y). That seems correct. Then substituting into the original equation. So (X, Y) = (8 - x, 2 - y). Then X² + Y² = 25 becomes (8 - x)^2 + (2 - y)^2 = 25. Expanding that gives x² -16x +64 + y² -4y +4 =25, which is x² + y² -16x -4y +68 =25, leading to x² + y² -16x -4y +43 =0. So yes, that seems correct.

Alternatively, if I consider that reflection is an invertible transformation, so the image should be another circle with the same radius, centered at (8,2). Therefore, expanding (x -8)^2 + (y -2)^2 =25 gives the same equation. So that's consistent.
</execution_4.1>
</Execution_4>

<Final_answer>
Therefore, the ordered quadruple $(a,b,c,d)$ of real numbers is $\boxed{(1, -16, -4, 43)}$.
</Final_answer>
</think>