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

Answers (1)

D Deependra Verma

Solution: We have ,

$fracx^2a^2+fracy^2a^2+w=1$ $........(1)$

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

$Rightarrow$ $frac2xa^2+frac2ya^2fracmathrmd ymathrmd x=0$ $........(2)$

$x+yfracmathrmd ymathrmd x=0$

$Rightarrow$ Now , to obtain orthogonal trajectory replace $fracmathrmd ymathrmd x$ by $-fracmathrmd xmathrmd y$

$Rightarrow$ $x+y(-fracmathrmd xmathrmd y)=0Rightarrow fracdyy=fracdxx$

Integrating both sides we get $ln y=ln x+ln cRightarrow y=xc$

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