JEE Main 2022 · 27 Jun · Shift 2 · Mathematics Q27

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MathematicsInteger+4 / -1Question 27 of 90

If the area of the region $\left\{(x,y):x^{2/3}+y^{2/3}\le1,\ x+y\ge0,\ y\ge0\right\}$ is A, then $\dfrac{256A}\pi$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 36

Astroid area. $x^{2/3}+y^{2/3}\le1$ has area $\frac{3\pi}8$ ; the upper half is $\frac{3\pi}{16}$ .

Cut by $x+y\ge0$ . In the second quadrant the line $y=-x$ meets the curve at $\left(-\frac1{2\sqrt2},\frac1{2\sqrt2}\right)$ ; removing the part below the line leaves $A=\frac{9\pi}{64}$ .

Value. $\dfrac{256A}\pi=36$ (confirmed numerically).

Answer: 36

Topics:Area between curves

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JEE Main 2022 · 27 Jun · Shift 2, Mathematics question 27