If the curves y4 = x and xy = k intersect at right angle, then the value of (4k)6 is

# If the curves  and  intersect at right angle, then the value of ${\left(4k\right)}^{6}$ is

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### Solution:

The equations of the curves are

${y}^{4}=x$ and $xy=k$

These two curves intersect at $\left({k}^{4/5},{k}^{1/5}\right).$

Now,

${y}^{4}=x$ and $xy=k$

and $x\frac{dy}{dx}+y=0$

and $\frac{dy}{dx}=-\frac{y}{x}$

and

If the curves intersect at right angle, then

$\frac{1}{4{k}^{3/5}}×-{k}^{-3/5}=-1⇒{k}^{6/5}=\frac{1}{4}⇒{k}^{6}={\left(\frac{1}{4}\right)}^{5}⇒\left(4k{\right)}^{6}=4$

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