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Let the circle C₂ be the mirror image of the circle C₁ : x^2 + y^2 - 6x - 8y + 16 = 0 in the line L : 3x - 4y + 17 = 0 . If C₁ and C₂ intersect at two distinct points P and Q , and A and B are the centers of C₁ and C₂ respectively, then the square of the area of the quadrilateral APBQ is ______

Correct answer

80

Step-by-step solution

The given circle is C₁ : x^2 + y^2 - 6x - 8y + 16 = 0 . The center of C₁ is A(3, 4) and its radius is r = 3^2 + 4^2 - 16 = 9 + 16 - 16 = 3 . Since C₂ is the mirror image of C₁ in the line L : 3x - 4y + 17 = 0 , the line L acts as the perpendicular bisector of the line segment joining the centers A and B . Also, the intersection points P and Q of the two circles lie on the line of reflection L , which is their common chord. Thus, the quadrilateral APBQ is a rhombus with diagonals AB and PQ . The distance d from the

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