Question

The focus of a conic is the origin and its corresponding directrix is $7x-y-10=0$. If length of its latus rectum is $2$, then its eccentricity is:

• A. $\sqrt{2}$
• B. $1$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{1}{2}$

Answer 1

The correct option is C $\frac{1}{\sqrt{2}}$

Perpendicular distance of directrix from focus is $\frac{|-10|}{5\sqrt{2}}=\sqrt{2}$
Given, length of latus rectum $=2$
Since, perpendicular distance from focus to directrix $\ne$ half of latus rectum.
Hence, the conic is not a parabola. So, $e\ne 1$
Now, let's assume it is an ellipse.
$L{L}^{\prime }=2=\frac{2b^2}{a}$
$\Rightarrow b^2=a$
Distance between focus and directrix is $\frac{a}{e}-ae=\sqrt{2}$
$\Rightarrow a(1-e^2)=e\sqrt{2}$
Also, for an ellipse, $b^2=a^2(1-e^2)$
$\Rightarrow a=ae\sqrt{2}$
$\Rightarrow e=\frac{1}{\sqrt{2}}$ ($\because a\ne 0$)
Since, $0<e<1$
Hence, our assumption is correct and $e=\frac{1}{\sqrt{2}}$