Let the line x - y = 4 intersect the circle C: (x-4)^2 + (y+3)^2 = 9 at the points Q and R . If P( , ) is a point on C such that PQ = PR , then (6 + 8 )^2 is equal to __________.

Step-by-step solution

The given circle C: (x-4)^2 + (y+3)^2 = 9 has its center at C₀(4, -3) and radius r = 3 . The line QR has the equation x - y = 4 , which has a slope of 1 . Since PQ = PR , the point P( , ) must lie on the perpendicular bisector of the chord QR . The perpendicular bisector of any chord of a circle passes through its center. Therefore, the perpendicular bisector passes through C₀(4, -3) and has a slope of -1 (since it is perpendicular to QR ). The equation of the perpendicular bisector is: y - (-3) = -1(x - 4) x + y =