Question

Chord of an ellipse are drawn through the positive end of the minor axis. Then their mid-point lies on

• A. a circle
• B. a parabola
• C. an ellipse
• D. a hyperbola

Answer 1

The correct option is C an ellipse

Let ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$ Then, the equation of the chord drawn from mid point $M(h,k)$ is
$T=S_1$
$\Rightarrow \frac{hx}{a^2}+\frac{ky}{b^2}-1=\frac{h^2}{a^2}+\frac{k^2}{b^2}-1\dots \left(1\right)$
Now, equation $\left(1\right)$ passing through $(0,b)$
$\frac{k}{b}=\frac{h^2}{a^2}+\frac{k^2}{b^2}$
Hence, the locus of $(h,k)$ is
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{y}{b}\frac{x^2}{a^2}+\frac{(y-b/2{)}^2}{b^2}=\frac{1}{4}$
which is an ellipse