The tangents PA and PB are drawn from any point P of the circle `x^(2)+y^(2)=2a^(2)` to the circle `x^(2)+y^(2)=a^(2)`. The chord of contact AB on extending meets again the first circle at the points A' and B'. The locus of the point of intersection of tangents at A' and B' may be given as

Options

A. `x^(2)+y^(2)=8a^(2)`
B. `x^(2)+y^(2)=4a^(2)`
C. `x^(2)+y^(2)=6a^(2)`
D. none of these

Answer

A

Explanation

The equation of the chord of contact of tangents drawn from any point `P(sqrt(2)a cos theta, sqrt(2) a cos theta)` to the circle `x^(2)+y^(2)=a^(2)` is
`x cos theta + y sin theta = (a)/(sqrt(2)) " " ...(i)`

Let Q(h, k) be the point of intersection of tangents to the circle `x^(2)+y^(2)=2a^(2)` at A' and B'. Then, the equation of the chord of contact A' B' is
`hx+ky =2a^(2) " " ...(ii)`
Clearly (i) and (ii) represent the same line.
`:. (cos theta)/(h) = (sin theta)/(k) = (1)/(2sqrt(2)a)`
`rArr cos theta = (h)/(2 sqrt(2)a), sin theta = (k)/(2sqrt(2)a)`
`rArr (h^(2))/(8a^(2))+(k^(2))/(8a^(2))=1 rArr h^(2)+k^(2)=8a^(2)`
Hence, the locus of (h, k) is `x^(2)+y^(2)=8a^(2)`