# Find the orthogonal trajectories of the family of the curves x^2/a^2+y^2/a^2+w=1 where w is parameter

Solution:  We have ,

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{a^{2}}+w=1$           $........(1)$

Diffrentiating $(1)$ with respect to $x$ , we have

$\Rightarrow$                      $\frac{2x}{a^{2}}+\frac{2y}{a^{2}}\frac{\mathrm{d} y}{\mathrm{d} x}=0$        $........(2)$

$x+y\frac{\mathrm{d} y}{\mathrm{d} x}=0$

$\Rightarrow$     Now , to  obtain orthogonal trajectory replace $\frac{\mathrm{d} y}{\mathrm{d} x}$ by $-\frac{\mathrm{d} x}{\mathrm{d} y}$

$\Rightarrow$         $x+y(-\frac{\mathrm{d} x}{\mathrm{d} y})=0\Rightarrow \frac{dy}{y}=\frac{dx}{x}$

Integrating both sides we get    $\ln y=\ln x+\ln c\Rightarrow y=xc$

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