Question

If the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is inscribed in a rectangle whose length to breadth ratio is $2:1,$ then the area of the rectangle is

• A. $4.\frac{(a^2+b^2)}{7}$
• B. $4.\frac{(a^2+b^2)}{3}$
• C. $12.\frac{(a^2+b^2)}{5}$
• D. $8.\frac{(a^2+b^2)}{5}$

Answer 1

Answer: (D)

$8.\frac{(a^2+b^2)}{5}$

$l=2a$

$b^{\prime }=2b$

$l:b^{\prime }=2:1\Rightarrow \frac{l}{b^{\prime }}=\frac{2}{1}\Rightarrow \frac{2a}{2b}=\frac{2}{1}\Rightarrow a=2b=b^{\prime }$

Area$=l\times b^{\prime }$ $=2a\times a$ $=2a^2$

OR

$=l\times b^{\prime }$

$=4b\times 2b$

$=8b^2$

Now using option D

Area$=\frac{8(a^2+b^2)}{5}=\frac{8(4b^2+b^2)}{5}=\frac{8\times 5b^2}{5}$

Area$=8b^2$

Answer is D.