Question

With a given point and line as focus and directrix, a series of ellipses are described. The locus of the extremities of their minor axis is

• A. an ellipse
• B. a parabola
• C. a hyperbola
• D. none of these

Answer 1

Answer: (B)

Let $S$ be the given focus and $ZM$ be the given line.
Then,

$SZ=\frac{a}{e}-ae$
$=\frac{a}{e}\left(1-e^2\right)=\frac{b^2}{ae}=k$ (say)
as $b^2=a^2\left(1-e^2\right)$
Now, take SC as the $x$-axis and LSL' as the $y$-axis. Let $(x,y)$ be the coordinates of $B$ with respect to these axes. Then, $x=SC=$ ae and $y=CB=b$.
Hence, $\frac{y^2}{x}=\frac{b^2}{ae}=SZ$ which is constant
Therefore, $y^2=kx$ is the required locus which is a parabola.