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 fouc as (2, 3)
• C. An ellipse with eccentricity
• D. An ellipse with length of latus rectrum is 2 units

Answer 1

The correct option is D An ellipse with length of latus rectrum is 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 $\Rightarrow r^2(l^2+m^2)=n^2\Rightarrow 1(x_1^2+4y_1^2)=16\Rightarrow x^2+4y^2=16isellipsewitheccentricitye=\frac{\sqrt{3}}{2}andlengthoflatusrectumL{L}^1=2$