JEE MAIN - Mathematics (2022 - 29th June Evening Shift - No. 15)
For real numbers a, b (a > b > 0), let
Area $$\left\{ {(x,y):{x^2} + {y^2} \le {a^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \ge 1} \right\} = 30\pi $$
and
Area $$\left\{ {(x,y):{x^2} + {y^2} \le {b^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \le 1} \right\} = 18\pi $$
Then, the value of (a $$-$$ b)2 is equal to ___________.
Answer
12
Explanation
$x^{2}+y^{2} \leq a^{2}$ is interior of circle
and $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \geq 1$ is exterior of ellipse
_29th_June_Evening_Shift_en_15_1.png)
$\therefore$ Area $=\pi a^{2}-\pi a b=30 \pi$
Similarly $x^{2}+y^{2} \geq b^{2}$ and $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \leq 1$ gives
$\pi a b-\pi b^{2}=18 \pi$
By (1) and (2), $\frac{a}{b}=\frac{5}{3} \Rightarrow a=\frac{5 b}{3}$
$\Rightarrow \pi \cdot \frac{25 b^{2}}{9}-\pi \cdot \frac{5 b^{2}}{3}=30 \pi$
$\Rightarrow\left(\frac{25}{9}-\frac{5}{3}\right) b^{2}=30$
$\Rightarrow \frac{10}{9} b^{2}=30 \Rightarrow b^{2}=27$
and $a^{2}=\frac{25}{9} \cdot 27=75$
$(a-b)^{2}=(5 \sqrt{3}-3 \sqrt{3})^{2}=3 \cdot 4=12$
and $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \geq 1$ is exterior of ellipse
$\therefore$ Area $=\pi a^{2}-\pi a b=30 \pi$
Similarly $x^{2}+y^{2} \geq b^{2}$ and $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \leq 1$ gives
$\pi a b-\pi b^{2}=18 \pi$
By (1) and (2), $\frac{a}{b}=\frac{5}{3} \Rightarrow a=\frac{5 b}{3}$
$\Rightarrow \pi \cdot \frac{25 b^{2}}{9}-\pi \cdot \frac{5 b^{2}}{3}=30 \pi$
$\Rightarrow\left(\frac{25}{9}-\frac{5}{3}\right) b^{2}=30$
$\Rightarrow \frac{10}{9} b^{2}=30 \Rightarrow b^{2}=27$
and $a^{2}=\frac{25}{9} \cdot 27=75$
$(a-b)^{2}=(5 \sqrt{3}-3 \sqrt{3})^{2}=3 \cdot 4=12$
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