Question

The locus of the points of intersection of the tangents at the extremities of the chords of the ellipse $x^2+2y^2=6$ which touch the ellipse $x^2+4y^2=4$ is

• A. $x^2+y^2=4$
• B. $x^2+y^2=6$
• C. $x^2+y^2=9$
• D. none of these

Answer 1

The correct option is C $x^2+y^2=9$

$\frac{x^2}{4}+y^2=1$ ... (1)

$\frac{x^2}{6}+\frac{y^2}{3}=1$ ... (2)

Now, the chords of ellipse (2) from points $P$ and $Q$ are tangents to ellipse (1).

Let the tangents at $P$ and $Q$ meet at $R(x_1,y_1)$.

Hence, we have $\frac{x{x}_1}{6}+\frac{y{y}_1}{3}=1$ as equation of tangent to ellipse (1).

Also, parametric form of tangent of ellipse (1) is $\frac{x\cos \theta }{2}+y\sin \theta =1$.

Since both these equations represent the same line, we compare to get

$\frac{\frac{x_1}{6}}{\frac{\cos \theta }{2}}=\frac{\frac{y_1}{3}}{\sin \theta }=1$

$⟹x_1=3\cos \theta$ and $y_1=3\sin \theta$ which is the parametric equation of circle $x_1^2+y_1^2=9$.

Hence, the locus is $x^2+y^2=9$.