Question

Chords of an ellipse are drawn through the positive end of the minor axis. Their midpoint lies on

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

Answer 1

Answer: (C)

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
$\frac{x_1+0}{2}=h\frac{y_1+b}{2}=k$
$\Rightarrow x_1=2h\Rightarrow y_1=2k-b$
$\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}=1\Rightarrow \frac{4h^2}{a^2}+\frac{(2k-b{)}^2}{b^2}=1$
$\Rightarrow \frac{h^2}{a^2/4}+\frac{{\left(k-\frac{b}{2}\right)}^2}{b^2/4}=1$
$\therefore \frac{x^2}{a^2/4}+\frac{{\left(y-\frac{b}{2}\right)}^2}{b^2/4}=1$
$\to \text{ Locus of midpoint which is an ellipse }$