If the line `xcostheta=2` is the equation of a transverse common tangent to the circles `x^2+y^2=4` and `x^2+y^2-6sqrt(3)x-6y+20=0` , then the value of `theta` is `(5pi)/6` (b) `(2pi)/3` (c) `pi/3` (d) `pi/6`

Options

A. `5pi //6`
B. `2pi //3`
C. `pi //3`
D. `pi//6`

Answer

3

Explanation

`C_(1)-=(0,0),C_(2)-=(3sqrt(3),3)`
and `r_(1)=2,r_(2)=4`
`:. C_(1)C_(2)=r_(1)+r_(2)`
Thus, circle touch each other externally .
The equation of common tangent at P is
`sqrt(3) x +y-4=0` (1)
Comparing it with ` r cos theta +y sin theta = 2` , we get `theta = pi //6`