Question

Equation of ellipse whose minor axis is equal to the distance between the foci and whose latus rectum is $10$, is given by (Take origin as centre and major axis along x-axis)

• A. $2x^2+y^2=100$
• B. $x^2+2y^2=100$
• C. $2x^2+y^2=50$
• D. none of these

Answer 1

Answer: (B)

$x^2+2y^2=100$

Let the equation of the ellipse is $dfrac{x}^2{a}^2+\frac{y^2}{b^2}=1a>b$

Then the foci are $S(ae,0)$ and $S^{\prime }(-ae,0)$

Length of minor axes $B{B}^{\prime }=2b$

Length of latus rectum $=\frac{2b^2}{a}$

According to the question

$B{B}^{\prime }=S{S}^{\prime }$

$2b=2ae$

$b=ae$

And

$\frac{2b^2}{a}=10$

$b^2=5a$

$a^2{e}^2=a^2(1-e^2)$

$e=\frac{1}{\sqrt{2}}$

$b=\frac{a}{\sqrt{2}}$

$b^2=\frac{a^2}{2}$

$5a=\frac{a^2}{2}$

$a=10$

$b^2=5\times 10=50$

Hence equation of ellipse

$\Rightarrow \frac{x^2+2y^2}{100}=1$

$\Rightarrow x^2+2y^2=100$