The orthogonal trajectory for the family of curves $y = k e^{x}$ is

e^{y} = - x + c

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y^{2} = - 2 x + c

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y^{2} = - x + c

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None of the above

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Solution

The correct option is B $y^{2} = - 2 x + c$
We have, $y = k e^{x}$ ....(1)
Differentiating w.r.t. $x$ , we get
$\frac{d y}{d x} = k e^{x} = y$ ....(2)
Replacing $\frac{d y}{d x}$ with $- \frac{d x}{d y}$ in (2), we get
$- \frac{d x}{d y} = y$
$\Rightarrow - y d y = d x$
On integrating, we get
$- \frac{1}{2} y^{2} = x + c / 2$
$\Rightarrow y^{2} = - 2 x + c$

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Similar questions

x^{2} + y^{2} = c

Find the orthogonal trajectory of $x^{2} + y^{2} = c$

y^{2} = c x^{3}

x + 1 = y + k e^{- y}

e^{x} + e^{- y} = k

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