Question

Let the variable line $y=kx+h$ is tangent to the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{9}=1.$ If the locus of $P\left(h , k\right)$ is a conic, then which of the following statement is false about this conic?

• A. Vertices of conic lies on y-axis
• B. The eccentricity of conic is greater than the eccentricity of given hyperbola.
• C. The centre of conic is at the origin
• D. The conic is an ellipse

Answer 1

Answer: (D)

Equation of the tangent $y=k x+h$ is compared with $y=m x \pm \sqrt{4 m^{2}-9}$
$\Rightarrow 1=\frac{m}{k}=\frac{\pm \sqrt{4 m^{2}-9}}{h}$
i.e. $m=k$ and $h=\sqrt{4 m^{2}-9}$
$\Rightarrow h=\sqrt{4 k^{2}-9}$
$\Rightarrow 4 y^{2}-x^{2}=9$
$\Rightarrow \frac{y^{2}}{\left(\frac{3}{2}\right)^{2}}-\frac{x^{2}}{3^{2}}=1$
$\Rightarrow 3^{2}=\left(\frac{3}{2}\right)^{2}\left( e ^{2}-1\right)$
$\Rightarrow e ^{2}=5 \Rightarrow e =\sqrt{5}>1$
Hence, the conic is not an ellipse