# If the curves x = y4 and xy = k cut at right angles, then $(4k)^6$ is equal to ______. Option: 1 4 Option: 2 6 Option: 3 8 Option: 4 12

Answers (1)

Given the equation of curves are

x = y4 and xy = k

$\\\text { for intersection } \quad \mathrm{y}^{5}=\mathrm{k} \ldots(1)\\ \text { Also } x=y^{4}\\ \Rightarrow 1=4 \mathrm{y}^{3} \frac{\mathrm{dy}}{\mathrm{dx}} \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{4 \mathrm{y}^{3}}$

$\\\text { for } \mathrm{xy}=\mathrm{k} \Rightarrow \mathrm{x}=\frac{\mathrm{k}}{\mathrm{y}} \\ \Rightarrow 1=-\frac{\mathrm{k}}{\mathrm{y}^{2}} \cdot \frac{\mathrm{dy}}{\mathrm{dx}} \\ \Rightarrow \frac{\mathrm{d} y}{\mathrm{dx}}=\frac{-\mathrm{y}^{2}}{\mathrm{k}}$

because Curve cut orthogonally

\begin{aligned} &\Rightarrow \frac{1}{4 y^{3}} \times\left(\frac{-y^{2}}{k}\right)=-1\\ &\Rightarrow y=\frac{1}{4 k}\\ &\therefore \text { from }(1) y^{5}=k\\ &\Rightarrow \frac{1}{(4 k)^{5}}=k\\ &\Rightarrow 4=(4 k)^{6} \end{aligned}

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