Question

A 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. $\frac{a^2}{x^2}-\frac{b^2}{y^2}=\frac{1}{2}$

Answer 1

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

Let equation of tangent to the given ellipse is $\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}=1$

Finding its intersection with X and Y axes

$A=(\frac{a^2}{x_1},0);B=(0,\frac{b^2}{y_1})$

Mid point is $(\frac{a^2}{2x_1},\frac{b^2}{2y_1})=(\alpha ,\beta )$

$(x_1,y_1)$ lies on Ellipse

$\Rightarrow \frac{a^2}{4{\alpha }^2}+\frac{b^2}{4{\beta }^2}=1$

$\therefore \frac{a^2}{x^2}+\frac{b^2}{y^2}=4$ is equation of required locus.