Question

If a chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ touches the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$, and the locus of middle point of such chord is given by ${(\frac{x^2}{4}+\frac{y^2}{9})}^{{\lambda }_2}={\lambda }_1(\frac{x^2}{4}-\frac{y^2}{9})$ then ${\lambda }_1+{\lambda }_2$ is $({\lambda }_1,{\lambda }_2\in N)$

• A. $4$
• B. $2$
• C. $1$
• D. $3$

Answer 1

The correct option is D $3$

Equation of $AB$

$(\frac{xh}{4}+\frac{yk}{9}-1)=\frac{h^2}{4}+\frac{k^2}{9}-1$ ... (i)
As (i) touches the hyperbola hence
$c^2=a^2{m}^2-b^2\Rightarrow \frac{81}{k^2}{(\frac{h^2}{4}+\frac{k^2}{9})}^2=\frac{81}{k^2}(\frac{h^2}{4}-\frac{k^2}{9})$