Tangents drawn from `P(1, 8)` to the circle `x^2 +y^2 - 6x-4y - 11=0` touches the circle at the points A and B, respectively. The radius of the circle which passes through the points of intersection of circles `x^2+y^2 -2x -6y + 6 = 0 and x^2 +y^2 -2x-6y+6=0` the circumcircle of the and interse `DeltaPAB` orthogonally is equal to

Options

A. `(sqrt(73))/(4)`
B. `(sqrt(71))/(2)`
C. 3
D. 2

Answer

A

Explanation

Equation of circle circumscribing `DeltaPAB` is:
`(x-1) (x-3) +(y-8) (y-2) =0`
`rArr x^(2)+y^(2) - 4x -10y +19 = 0` (i)
Equation of circle passing through points of intersection of circles `x^(2) +y^(2) -2x -6y +6 = 0` and `x^(2) +y^(2) +2x - 6y +6 = 0` is given by
`(x^(2) +y^(2) -2x -6y +6) + lambda (x^(2) +y^(2) +2x -6y +6) =0`
`rArr x^(2)+ y^(2) +((2lambda -2))/(lambda+1) x - 6y +6 = 0` (ii)
As circle (ii) is orthogonal to circle (i), we have
`-2 ((2lambda-2)/(lambda+1)) - 5 (-6) = 19+6`
`rArr 4 lambda -4 = 5 lambda +5`
`rArr lambda =- 9`
Hence, required equation of circle is:
`x^(2) +y^(2) +(5)/(2) x -6y +6 = 0`
`:.` Radius of circle `= sqrt((25)/(16)+9-6) = (sqrt(73))/(4)`