In ` A B C` , the coordinates of the vertex `A` are `(4,-1)` , and lines `x-y-1=0` and `2x-y=3` are the internal bisectors of angles `Ba n dC` . Then, the radius of the encircle of triangle `A B C` is `4/(sqrt(5))` (b) `3/(sqrt(5))` (c) `6/(sqrt(5))` (d) `7/(sqrt(5))`

Options

A. `4//sqrt(5)`
B. `3//sqrt(5)`
C. `6//sqrt(5)`
D. `7//sqrt(5)`

Answer

C

Explanation

Incentre I is point of intersection of two angle bisectors which is (2,1).
`IA = sqrt(2^(2) + 2^(2)) = 2 sqrt(2)`
`"Also, AI" = r " cosec" (A)/(2)`
`angleBIC = (pi)/(2) + (A)/(2)`
`"or tan"((pi)/(2)+(A)/(2)) = |(1-2)/(1+2)|`
`"or tan "(A)/(2) = 3`
`"So, r" = (AI)/("cosec"(A)/(2)) = (2sqrt(2))/(sqrt(1+(1)/(9))) (2sqrt(2) xx 3)/(sqrt(10)) = (6)/(sqrt(5))`