Question

The extremities of a diagonal of a rectangle are (4, 4) and (6, 1). A circle circumscribes the rectangle and cuts an intercept AB on the y-axis. Then the area of the triangle formed by AB and the two tangents at A and B is

(a)  (11/2)²

(b)  (11/2)²

(c)   (11/2)³

(d)  (11/2)⁴

Answer 1

Correct option (c)   (11/2)³

Explanation :

The circumcircle of the rectangle is

(x + 4)(x - 6) + (y - 4)(y + 1) = 0

x + y - 2x - 3y - 28 = 0....(1)

The circumcircle of the rectangle is  the length of the y-intercept of the circle provided in Eq. (1) is

Therefore, AB = 11. The ordinates of A and B are given by

Therefore, A = (0,7 ) and B = (0 , -4). 07 0 4 Equation of the tangent at A(0, 7) is

Equation of the tangent at B(0, 4) is

Solving Eqs. (2) and (3), we have C = ( 121/4, 3/2) which is the intersection of the tangents at A and B. Therefore, area of ΔABC is