Question

If a variable tangent of the circle $x^2+y^2=1$ intersect the ellipse $x^2+2y^2=4$ at P and Q then the locus of the points of intersection of the tangents at P and Q is

• A. a circle of radius 2 units
• B. a parabola with focus as (2, 3)
• C. an ellipse with eccentricity $\frac{\sqrt{3}}{4}$
• D. an ellipse with length of latusrectum 2 units

Answer 1

The correct option is D an ellipse with length of latusrectum 2 units

$x^2+y^2=1;x^2+2y^2=4$
Let $R(x_1,y_1)$ is point of intersection of tangents drawn at P,Q to ellipse
$\Rightarrow$ PQ is chord of contact of $R(x_1,y_1)$
$\Rightarrow x{x}_1+2y{y}_1-4=0$
This touches circle $\Rightarrow r^2(l^2+m^2)=n^2$
$\Rightarrow 1(x_1^2+4y_1^2)=16$
$\Rightarrow x^2+14y^2=16$ is ellipse $e=\frac{\sqrt{3}}{2};L{L}^{\prime }=2$.