Question

For an ellipse, the locus of mid points of chords which are drawn through an end of minor axis will be

• A. a parabola
• B. an ellipse
• C. a hyperbola
• D. a pair of lines

Answer 1

Answer: (B)

Let the equation of the given ellipse be $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ .
Let $\left(a cos \theta , b sin \theta \right)$ be the co-ordinates of the other extremities of the chord of the ellipse.
So, the positive end of the minor axis is $\left(0 , b\right)$ .
Let $\left(h , k\right)$ be the mid-point of the chord.
So, $h=\frac{a cos \theta + 0}{2}\Rightarrow cos\theta =\frac{2 h}{a}..........\left(i\right)$
and $k=\frac{b sin \theta + b}{2}\Rightarrow sin\theta =\frac{2 k - b}{b}..........\left(i i\right)$
Now, squaring and adding both the equations, we get
$\left(sin\right)^{2}\theta +\left(cos\right)^{2}\theta =\left(\frac{2 k - b}{b}\right)^{2}+\left(\frac{2 h}{a}\right)^{2}$
$\Rightarrow \frac{4 k^{2} + b^{2} - 4 k b}{b^{2}}+\frac{4 h^{2}}{a^{2}}=1$
$\Rightarrow \frac{4 k^{2} - 4 k b}{b^{2}}+1+\frac{4 h^{2}}{a^{2}}=1$
$\Rightarrow \frac{4 k^{2} - 4 k b}{b^{2}}+\frac{4 h^{2}}{a^{2}}=0$
$\Rightarrow \frac{h^{2}}{a^{2}}+\frac{k^{2}}{b^{2}}-\frac{k}{b}=0$
For locus, replacing $h \rightarrow x\&k \rightarrow y$ , we get
$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{y}{b}=0$ which is an ellipse
So, the required locus is an ellipse.