JEE PYQ: Application Of Derivatives Question 20
Question 20 - 2021 (25 Feb Shift 2)
If the curves $x = y^4$ and $xy = k$ cut at right angles, then $(4k)^6$ is equal to
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Answer: 4
Solution
From $x = y^4$: $\frac{dy}{dx} = \frac{1}{4y^3}$. From $xy = k$: $\frac{dy}{dx} = \frac{-y}{x}$. Orthogonality: $\frac{1}{4y^3} \cdot \frac{-y}{x} = -1$. With $x = y^4$ and $xy = k$: $y^6 = \frac{1}{4}$, $k^6 = y^{30} = \left(\frac{1}{4}\right)^5$. So $(4k)^6 = 4^6 \cdot k^6 = 4^6 \cdot \frac{1}{4^5} = 4$.
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