JEE Main 2022 · 24 Jun · Shift 1 · Mathematics Q23

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MathematicsInteger+4 / -1Question 23 of 90

Let $A\left(\dfrac3{\sqrt a},\sqrt a\right)$ , $a>0$ , be a fixed point in the $xy$ -plane. The image of $A$ in $y$ -axis be $B$ and the image of $B$ in $x$ -axis be $C$ . If $D(3\cos\theta,a\sin\theta)$ is a point in the fourth quadrant such that the maximum area of $\Delta ACD$ is 12 square units, then $a$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 8

Points. $B=\left(-\frac3{\sqrt a},\sqrt a\right)$ and $C=\left(-\frac3{\sqrt a},-\sqrt a\right)=-A$ .

Area. Since $C=-A$ , the triangle $ACD$ has area $|A\times D|=\left|\frac3{\sqrt a}\cdot a\sin\theta-\sqrt a\cdot3\cos\theta\right|=3\sqrt a\,|\sin\theta-\cos\theta|$ .

Maximum in the fourth quadrant ( $\sin\theta<0<\cos\theta$ ): $|\sin\theta-\cos\theta|=\sqrt2\left|\sin\left(\theta-\frac\pi4\right)\right|$ reaches $\sqrt2$ at $\theta=-\frac\pi4$ .

Solve. $3\sqrt{2a}=12$ , so $a=8$ .

Answer: 8

Topics:Ellipse Distance and section formulae AM-GM-HM inequalities

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JEE Main 2022 · 24 Jun · Shift 1, Mathematics question 23