Let `P(alpha,beta)` be a point in the first quadrant. Circles are drawn through P touching the coordinate axes.
Radius of one of the circles is

Options

A. `(sqrt(a)-sqrt(beta))^(2)`
B. `(sqrt(alpha)+sqrt(beta))^(2)`
C. `alpha +beta -sqrt(alpha beta)`
D. `alpha +beta -sqrt(2alpha beta)`

Answer

D

Explanation

Let equation of circle be `(x-r)^(2) + (y-r)^(2) = r^(2)`
`rArr x^(2) +y^(2) -2r (x+y) +r^(2) =0`, where r is radius of circle which is passing through `(alpha, beta)` so
`alpha^(2) + beta^(2) -2r (alpha +beta) +r^(2) =0`
`rArr r = (2(alpha+beta)+-sqrt(4(alpha+beta)^(2)-4(alpha^(2)+beta^(2))))/(2)`
`rArr r = (alpha + beta) +- sqrt(2 alpha beta)`