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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.

Accepted Answer

The center of the original circle is (0,0). The reflection of the point (0,0) in the point (4,1) is (8,2), so the equation of the new circle is (x - 8)^2 + (y - 2)^2 = 25.This simplifies to x^2 + y^2 - 16x - 4y + 43 = 0. Hence, (a,b,c,d) = (1,-16,-4,43).