Question

The chord of an ellipse are drawn through the positive end of the minor axis. Then, the mid-point lies on

• A. A circle
• B. A parabola
• C. An ellipse
• D. A hyperbola

Answer 1

Answer: (C)

An ellipse

Let $(h,k)$ be the mid-point of a chord passing through the positive end of the minor axis of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Then, the equation of the chord is
$\frac{hx}{a^2}+\frac{ky}{b^2}-1=\frac{h^2}{a^2}+\frac{k^2}{b^2}-1(\because T=S_1)$
$\Rightarrow \frac{hx}{a^2}+\frac{ky}{b^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}$
It passes through $(0,b)$
$\therefore 0+\frac{kb}{b^2}=\frac{h^2}{a^2}+\frac{k^2}{b^2}\Rightarrow \frac{k}{b}=\frac{h^2}{a^2}+\frac{k^2}{b^2}$
Hence, locus of a point is $\frac{y}{b}=\frac{x^2}{a^2}+\frac{y^2}{b^2}$
which represents an equation of ellipse.