Question

If tangent to $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the axes at $A$ and $B,$ then the locus of mid point of $AB$ is

• A. $\frac{a^2}{x^2}+\frac{b^2}{y^2}=2$
• B. $\frac{a^2}{x^2}+\frac{b^2}{y^2}=4$
• C. $\frac{a^2}{x^2}+\frac{b^2}{y^2}=1$
• D. none of these

Answer 1

The correct option is B $\frac{a^2}{x^2}+\frac{b^2}{y^2}=4$

Let the equation of tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be $\frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1$
So, the points
$A=(\frac{a}{\cos \theta },0),B=(0,\frac{b}{\sin \theta })$
Let the mid point of $AB$ be $(h,k)$
$\Rightarrow \frac{a}{2\cos \theta }=h,\frac{b}{2\sin \theta }=k$
$\Rightarrow 2\cos \theta =\frac{a}{h},2\sin \theta =\frac{b}{k}$
$\therefore$ The required locus is $\frac{a^2}{x^2}+\frac{b^2}{y^2}=4$