Let A B C be the triangle with A B = 1 , A C = 3 and ∠ B A C = π 2 . If a circle of radius r > 0 touches the sides A B , A C and also touches internally the circumcircle of the triangle A B C , then the value of r is ______. (round-off the value to two decimal place)
A. A
Let A be the origin B on x - axis, C on y - axis, So on plotting the diagram we get, Now equation of circumcircle is given by x - 1 2 2 + y - 3 2 2 = 1 2 2 + 3 2 2 = 5 2             ⋯ 1 Required equation of circle touches A B and A C and have radius r so its equation is given by, x - r 2 + y - r 2 = r 2           ⋯ 2 Now given circle in equation 2 touches circumcircle internally, so by property distance between the centre will be equal to modulus of difference of radii, So, d c 1 c 2 = r 1 - r 2 ⇒ 1 2 - r 2 + 3 2 - r 2 = 5 2 - r 2 ⇒ 1 4 + r 2 - r + 9 4 + r 2 - 3 r = 5 2 - r 2 ⇒ 2 r 2 - 4 r + 5 2 = 5 2 + r 2 - 10 r ⇒     r = 0 or r = 4 - 10 = 0 . 837 So, r = 0 . 84 (on rounding off)
Related: Mathematics — Properties of Triangles · All PYQ Banks