Question

The ratios of the areas of the curves $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{a^2}+\frac{y^2}{b^2}\le 1\le \frac{x}{a}+\frac{y}{b}$ is:

• A. $3:2$
• B. $4\pi :\pi -2$
• C. $\pi -2:4\pi$
• D. $\pi -2:\pi +2$

Answer 1

The correct option is B $4\pi :\pi -2$

curve $1$ is ellipse having semi major axis =$a$

and semi minor axis =$b$

hence $A^{\ast }=\pi ab$ --------equ$\left(1\right)$

the area bounded by the ellipse having semi major axis =$a$ and semi minor axis =$b$ and line intersect the$X$ - axis at $a$ and $Y$- axis at $b$ is given in figure

now $A^{@}=A_1-A_2$

$A_1=\pi ab$ and$A_2=\frac{ab}{2}$

now $A^{@}=\pi ab-\frac{ab}{2}$ ------equ($2$)

on dividing equ($1$) and equ($2$

)

we will get $\frac{A^{\ast }}{A^{@}}=\frac{4\pi }{\pi -2}$