A circle with radius `|a|` and center on the y-axis slied along it and a variable line through (a, 0) cuts the circle at points `Pa n dQ` . The region in which the point of intersection of the tangents to the circle at points `P` and `Q` lies is represented by (a)`y^2geq4(a x-a^2)` (b) `y^2lt=4(a x-a^2)` (c)`ygeq4(a x-a^2)` (d) `ylt=4(a x-a^2)`

Options

A. `y^(2) ge 4a (x-a)`
B. `y^(2) le 4ax`
C. `x^(2) +y^(2) le 4a^(2)`
D. `x^(2) -y^(2) ge a^(2)`

Answer

A

Explanation

Let the circle be `x^(2) + (y-alpha)^(2) =a^(2)`. Let the point of intersection of tangents at P and Q be (h,k).
Then equation of PQ, is `hx + (k-alpha) (y-alpha) -a^(2) =0`.
As PQ passes through (a,0), we have
`ha -alpha (k-alpha) -a^(2) =0`
`rArr alpha^(2) -k alpha +a (h-a) =0`
Now for real values of `alpha`, we have `D ge 0`.
`rArr k^(2) -4a(h-a) ge 0`
or `y^(2) ge 4a (x-a)`