Orthogonal trajectory of family of curve x23+y23=a where a is arbitrary constant is

23x23−1−23y23−1dydx=0Now replace dydx by dxdy and then solve the differential equation to get orthogonal curve

Orthogonal trajectory of family of curve x23+y23=a where a is arbitrary constant is

23x23−1−23y23−1dydx=0Now replace dydx by dxdy and then solve the differential equation to get orthogonal curve

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Orthogonal trajectory of family of curve $x^{\frac{2}{3}} + y^{\frac{2}{3}} = a$ where $a$ is arbitrary constant is

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$x^{\frac{2}{3}} - y^{\frac{2}{3}} = c$

$x^{\frac{2}{3}} + y^{\frac{2}{3}} = c x$

$x^{\frac{4}{3}} - y^{\frac{4}{3}} = c$

$x^{\frac{4}{3}} - y^{\frac{4}{3}} = c x$

answer is C.

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Detailed Solution

$\frac{2}{3} x^{\frac{2}{3} - 1} - \frac{2}{3} y^{\frac{2}{3} - 1} \frac{d y}{d x} = 0$

Now replace $\frac{d y}{d x}$ by $\frac{d x}{d y}$ and then solve the differential equation to get orthogonal curve

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Orthogonal trajectory of family of curve x23+y23=a where a is arbitrary constant is

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Orthogonal trajectory of family of curve $x^{\frac{2}{3}} + y^{\frac{2}{3}} = a$ where $a$ is arbitrary constant is