Let y=x be the equation of a chord of the circle C ₁ (in the closed half-plane x 0 ) of diameter 10 passing through the origin. Let C ₂ be another circle described on the given chord as its diameter. If the equation of the chord of the circle C ₂ , which passes through the point (2,3) and is farthest from the center of C ₂ , is x+a y+b=0 , then a-b is equal to

Step-by-step solution

Circle C₁ has radius 5 and chord on line y = x through origin. Setting center at (5, 0) (in region x 0 ), the chord endpoints are (0, 0) and (5, 5) by solving (t-5)^2 + t^2 = 25 . Circle C₂ has this chord as diameter, so center is at ( 5 2 , 5 2 ) with radius 5 2 2 . The chord of C₂ through (2, 3) that is farthest from center is perpendicular to the radial direction from center to (2, 3) . The radial direction is (2 - 5 2 , 3 - 5 2 ) = (- 1 2 , 1 2 ) with normal direction (1, -1) . The chord equation is 1(x - 2) -