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Q.
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
Conic Sections
Solution:
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, $\quad x=S C=$ ae and $y=C B=b$.
Hence, $\frac{ y ^{2}}{ x }=\frac{ b ^{2}}{ ae }= S Z $ which is constant
Therefore, $y^{2}=k x$ is the required locus which is a parabola.
