Question

The locus of mid-point of a focal chord of the ellipse $is\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is

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

Answer 1

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

To find the locus of mid-point of a focal chord of the ellipse

Equation of ellipse is

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Let $M(x_1,y_1)$ be the mid point of the chord

So , Equation of this chord is

$\frac{x{x}_1}{a^2}+\frac{y{y}_1}{b^2}=\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}$

If this chord is the focal chord of the ellipse , then

it must pass through the focus

$\frac{ae.x_1}{a^2}+0=\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}$

or

$\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}=\frac{e.x_1}{a}$

Hence, Locus of the Point $M(x_1,y_1)$ is

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{ex}{a}$

Option A is correct.