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 \quad \frac{y_1+b}{2}=k $
$ \Rightarrow x_1=2 h \Rightarrow y_1=2 k-b $
$ \frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}=1 \Rightarrow \frac{4 h^2}{a^2}+\frac{(2 k-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$
$ \rightarrow \text { Locus of midpoint which is an ellipse }$