JEE PYQ: Application Of Derivatives Question 49
Question 49 - 2020 (08 Jan Shift 1)
For $a > 0$, let the curves $C_1: y^2 = ax$ and $C_2: x^2 = ay$ intersect at origin $O$ and a point $P$. Let the line $x = b$ $(0 < b < a)$ intersect the chord $OP$ and the $x$-axis at points $Q$ and $R$, respectively. If the line $x = b$ bisects the area bounded by the curves $C_1$ and $C_2$, and the area of $\triangle OQR = \frac{1}{2}$, then $a$ satisfies the equation:
(1) $x^6 - 6x^3 + 4 = 0$
(2) $x^6 - 12x^3 + 4 = 0$
(3) $x^6 + 6x^3 - 4 = 0$
(4) $x^6 - 12x^3 - 4 = 0$
Show Answer
Answer: (2)
Solution
Curves meet at $(a, a)$. Area of $\triangle OQR = \frac{1}{2}b^2 = \frac{1}{2} \Rightarrow b = 1$. Bisection condition with area computation gives $4a\sqrt{a} - 2 = a^3$, leading to $a^6 - 12a^3 + 4 = 0$.
BETA
