If a chord of contact of tangents drawn from a point P with respect to the circle `x^(2)+y^(2)=9` is x=2, then area, in square units, of triangle formed by tangents drawn from P to the circle and their chord of contact is equal to

Options

A. `(4 sqrt(5))/(2)`
B. `(9sqrt(3))/(2)`
C. `5sqrt(5))/(2)`
D. none of these

Answer

C

Explanation

Let `(x_(1),y_(1))` be the coordinates of P. Then, the equation of AB is `x x_(1)+y y_(1)=9`.
Clearly, this equation and x=2 represent the same line
`:. (x_(1))/(1)=(y_(1))/(0)=(9)/(2)rArr x_(1)=(9)/(2), y_(1)=0`.
We know that the area of the triangle formed by the tangents drawn from a point `P(x_(1), y_(1))` to the circle `x^(2)+y^(2)=a^(2)` and their chord of contact is
`(a(x_(1)^(2)+y_(1)^(2)-a^(2))^(3//2))/(x_(1)^(2)+y_(1)^(2))`
`:. ` Required area `=(3((81)/(4)+0-9)^(3//3))/((81)/(4)+0)` sq. units`=(5sqrt(5))/(2)`sq. units