Let the tangent to the circle x 2 + y 2 = 25 at the point R ( 3 , 4 ) meet x -axis and y -axis at point P and Q , respectively. If r is the radius of the circle passing through the origin O and having centre at the incentre of the triangle O P Q , then r 2 is equal to

Step-by-step solution

The equation of tangent to a circle x 2 + y 2 = a 2 at a point x 1 ,   y 1 is x x 1 + y y 1 = a 2 . Thus, the tangent to circle x 2 + y 2 = 25 at R 3 ,   4 is 3 x + 4 y = 25 . To find the coordinates of the point P on the x -axis, put y = 0 , to get 3 x = 25 ⇒ x = 25 3 Thus, the coordinates of the point P are 25 3 ,   0 . Similarly, to find the coordinates of the point Q on the y -axis, put x = 0 , to get 4 y = 25 ⇒ y = 25 4 Thus, the coordinates of the point Q are 0 ,   25 4 . The i