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
A. -2
Circle C_1 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 C_2 has this chord as diameter, so center is at ( 5 2 , 5 2 ) with radius 5 2 2 . The chord of C_2 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) - 1(y - 3) = 0, giving x - y + 1 = 0. In form x + ay + b = 0, we have a = -1 and b = 1, so a - b = -1 - 1 = -2.
Related: Mathematics — Circle · All PYQ Banks