Question

$S$ is one focus of an standard horizontal ellipse and $P$ is any point on the ellipse. If the maximum and minimum values of $SP$ are $m$ and $n$ respectively, then the length of semi minor axis is

• A. $AM$ of $m,n$
• B. $GM$ of $m,n$
• C. $HM$ of $m,n$
• D. none of these

Answer 1

The correct option is B $GM$ of $m,n$

Let the equation of the ellipse is
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Let focal distance of point $P=SP$ be$=a-ex$
If $P$ has coordinates, $(a\cos \theta ,b\sin \theta )$
then $SP=a-ea\cos \theta$
$SP=a(1-e\cos \theta )$
$\therefore S{P}_{max}=a(1+e)=m$
and, $S{P}_{min}=a(1-e)=n$
Now, consider
$m\cdot n=a^2(1-e^2)=b^2$
$b=\sqrt{mn}$
Clearly, semi minor axis is $GM$ of $m,n$