If the polar of a point ( p,q) with respect to the circle `x^(2)+ y^(2) =a^(2)` touches the circle `( x-c)^(2)+(y-d)^(2) +b^(2) `,then `:`

Options

A. `b^(2)(p^(2)+q^(2))=(a^(2)-cp-dq)^(2)`
B. `b^(2)(p^(2)+q^(2))=(a^(2)-cq-dp)^(2)`
C. `a^(2)(p^(2)+q^(2))=(b^(2)-cp-dq)^(2)`
D. none of these

Answer

A

Explanation

The polar of point (p, q) with respect of the circle `x^(2)+y^(2)=a^(2)` is
`px+qy=a^(2)`
This line touches `(x-c)^(2)+(y-d)^(2)=b^(2)`
`:. |(cp+dq-a^(2))/(sqrt(p^(2)+q^(2)))|=b rArr (a^(2)-cp-dq)^(2)=b^(2)(p^(2)+q^(2))`