Question

If the length of the major axis of a variable ellipse is constant, while that of the minor axis is variable, then the locus of one of the end points of its latus rectum, is a

• A. straight line
• B. circle
• C. parabola
• D. hyperbola

Answer 1

Answer: (C)

parabola

The length of the major axis, $2a$ is constant.
The length of the minor axis $2b$ is variable.
We have, $b^2=a^2(1-e^2)$.
One of the end points of the latus rectum$=(ae,b^2/a)=(h,k)$.
$h=ae$
$k=a(1-e^2)$
Eliminating $e$ from the two equations we get,
$\frac{h^2}{a^2}=1-\frac{k}{a}$
$\Rightarrow x^2=a(a-y)$
Hence, the locus of one of the end points of the latus rectum is a parabola.