Question

The locus of middle points of the chords of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ which passes through the positive end of the major axis is

• A. $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{y}{2b}=0$
• B. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{x}{a}$
• C. $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{x}{a}=0$
• D. $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{2x}{a}=0$

Answer 1

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

Equation of chord with mid point $(x,y)$ is
$\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}=\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}$
As it passes through $(a,0)$
$\Rightarrow \frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}=\frac{a{x}_1}{a^2}$
$\Rightarrow \frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}=\frac{x_1}{a}$, which is required locus.