A tangent to the circle `x^(2)+y^(2)=1` through the point (0, 5) cuts the circle `x^(2)+y^(2)=4` at P and Q. If the tangents to the circle `x^(2)+y^(2)=4` at P and Q meet at R, then the coordinates of R are

Options

A. `(8sqrt( 6)//5, 4//5)`
B. `(8sqrt(6)//5), -4//5)`
C. `(-8sqrt(6)//5), -4//5)`
D. none of these

Answer

A

Explanation

Clearly, PQ is the chord of contact of tangents drawn from `R(alpha, beta)` , to the circle `x^(2)+y^(2)=4`. So, its equation is
`alpha x + beta y = 4 " " ...(i)`
It passes through (0, 5) . Therefore, `beta=(4)/(5)`

Since PQ touches `x^(2)+y^(2)=1`.
`:. (4)/(sqrt(alpha^(2)+beta^(2)))=1 rArr alpha^(2)+beta^(2)=16 rArr alpha^(2)=16-(16)/(25) rArr alpha= pm (8sqrt(6))/(5)`
Thus, the coordinates of R the `(pm 8 sqrt(6)//5, 4//5)`