For real numbers a, b(a > b > 0) , let Area \ (x, y): x ^ 2 + y ^ 2 ≤ a ^ 2 ) and x ^ 2 a ^ 2 + y ^ 2 b ^ 2 ≥ 1\ = 30π and Area \ (x, y): x ^ 2 + y ^ 2 ≥ b ^ 2 ) and x ^ 2 a ^ 2 + y ^ 2 b ^ 2 ≤ 1\ = 18π Then the value

JEE Main 202229 JuneEvening Shift

Exam: JEE Main
Year: 2022
Date: 29 June
Shift: Evening Shift
Subject: Mathematics
Chapter: Area Under Curves

JEE Main 2022 · 29 June · Evening Shift · Mathematics · Area Under Curves

For real numbers a, b(a > b > 0) , let Area \ (x, y): x ^ 2 + y ^ 2...

For real numbers $a, b\left(a > b > 0\right)$ , let

Area \left\{\left(x, y\right): {x}^{2} + {y}^{2} \le {a}^{2}\right) $and \left\dfrac{{x}^{2}}{{a}^{2}} + \dfrac{{y}^{2}}{{b}^{2}} \ge 1\right\} = 30\pi$

and

Area \left\{\left(x, y\right): {x}^{2} + {y}^{2} \ge {b}^{2}\right) $and \left\dfrac{{x}^{2}}{{a}^{2}} + \dfrac{{y}^{2}}{{b}^{2}} \le 1\right\} = 18\pi$

Then the value of \left(a - b\right)2 is equal to _____.

Correct answer

Step-by-step solution

We know that area of circle $= {\pi r}^{2}$ ( $r$ is radius of circle) and area of ellipse $= \pi ab$ (where $a \text{ and } b$ are length of semi major and semi minor axis)

Now given area between circle and ellipse $⇒ \pi{a}^{2} - \piab = 30\pi$ .....(i)

And $⇒ \piab - \pi{b}^{2} = 18\pi$ ....(ii)

On subtracting equation (i) - (ii), we get ${\left(a - b\right)}^{2} = {a}^{2} - 2ab + {b}^{2} = 12$ .

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