Question

Find the equation of the ellipse under given conditions

Length of Latus Rectum $=10,$ distance between foci $=$ length of minor axis.

• A. $x^2+2y^2=100$
• B. $x^2+4y^2=400$
• C. $8x^2+5y^2=4000$
• D. None of these

Answer 1

The correct option is A $x^2+2y^2=100$

Given $\left(\frac{b^2}{a}\right)=10$ and $2ae=2b\Rightarrow b=ae$
Also we know that $b^2=a^2-a^2{e}^2$ or $a^2{e}^2=a^2-a^2{e}^2$ or $2e^2=1$
$\therefore e=\frac{1}{\sqrt{2}}$
$\frac{2b^2}{a}=10$ or $\frac{2a^2(1-e^2)}{a}=10$ or $a(1-\frac{1}{2})=5\therefore a=10$
and $b=ae=10\cdot \frac{1}{\sqrt{2}}=5\sqrt{2}.$
Hence $a^2>b^2,$ the equation of the ellipse is $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
or $\frac{x^2}{100}+\frac{y^2}{50}=1$ $\Rightarrow x^2+2y^2=100$