If the curves 2x = y2 and 2xy = k, intersect orthogonally, then the value of k2 =

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If the curves $2 x = y^{2}$ and $2 x y = k$ , intersect orthogonally, then the value of $k^{2} =$

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Detailed Solution

Given equations of curves are
$2 x = y^{2} ... (i)$
And $2 x y = k ... (i i)$
Since curves intersect orthogonally,
${(\frac{d y}{d x})}_{1} {(\frac{d y}{d x})}_{2} = (\frac{1}{y}) (- \frac{y}{x}) = - 1$ or $x = 1$
So, from (i), $y = \pm \sqrt{2}$
Putting these values in equation (ii), we get
$k = \pm 2 \sqrt{2}$