Go To Your Study Dashboard

Get Answers to all your Questions

header-bg

Find the orthogonal trajectories of the family of curves: (x^2/a^2)+y^2/(a^2+w)=1

Answers (1)

best_answer

Solution: Given ,

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

Diffrentiating $(1)$ , w.r.t $x$ ,

$Rightarrow$ $frac2xa^2+frac2ya^2+wfracmathrmd ymathrmd x=0Rightarrow frac1a^2+w=-fracxa^2yfracmathrmd xmathrmd y$ $......(2)$

Eliminating w fram $(1)$ and $(2)$ , we get

$Rightarrow$ $fracx^2a^2+y^2(-fracxa^2yfracmathrmd xmathrmd y)=1$ or $fracx^2a^2-fracxya^2frac1fracmathrmd ymathrmd x=1,$ $......(3)$

Which is the diffrential equation of the given family of curve $(1)$ .

Replacing $fracmathrmd ymathrmd x$ by $-fracmathrmd xmathrmd y$ in $(3)$ the differential equation of the orthiogonal trajectories is

$Rightarrow$ $fracx^2a^2-fracxya^2(-frac1fracmathrmd xmathrmd y)=1$ or $fracx^2a^2-fracxya^2fracmathrmd ymathrmd x=1$

$Rightarrow$ $fracxya^2fracmathrmd ymathrmd x=fraca^2-x^2a^2$ or $ydy=[fraca^2x-x]dx$

Integrating , $fracy^22=a^2log x-fracx^22+fracc2Rightarrow x^2+y^2-2a^2log x=c.$

Posted by

Deependra Verma

View full answer

Similar Questions

Latest Question