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