If a variable tangent of the circle $x^{2} + y^{2} = 1$ intersects the ellipse $x^{2} + {2y}^{2} = 4$ at points $P$ and $Q$, then the locus of the point of intersection of tangent at $P$ and $Q$ is

Question

Dr. Abubacker · Accepted Answer

(c)
Let point of intersection be $(h,\ k)$. Then, equation of the line passing through $P$ and $Q$ is $hx + 2ky = 4$ (chord of contact)
Since, $hx + 2ky = 4$ touches $x^{2} + y^{2} = 1,\ \frac{16}{4k^{2}} = 1 + \frac{h^{2}}{4k^{2}}$
$ie,\ 4k^{2} + h^{2} = 16$. So, required locus is $4y^{2} + x^{2} = 16$, which is an ellipse of eccentricity $\frac{\sqrt{3}}{2}$ and length of latusrectum is 2 unit

Q. If a variable tangent of the circle $x^{2} + y^{2} = 1$ intersects the ellipse $x^{2} + {2y}^{2} = 4$ at points $P$ and $Q$, then the locus of the point of intersection of tangent at $P$ and $Q$ is

Explanation

Reviewed by Dr. Abubacker

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