find equation of orthogonal trajectories of 1parameter famly of elipse 1/2x^2+y^2=C

Question

orthogonal trajectories , where C>0 value of C coresspond one of ther elipses

Answer 1

Question: find equation of orthogonal trajectories of 1parameter famly of elipse 1/2x^2+y^2=C.

The family of curves is,

1/2x^2 + y^2 = C

differentiate wrt x,

x + 2y.y' = 0

y' = -x/2y

Here y' gives the slope of the current family of curves.

Then -1/y' (i.e. 2y/x) is the slope of the orthogonal family of curves.

A DE describing the orthogonal family of curves is given by

y' = 2y/x

dy/dx = 2y/x

int (1/y) dy = int (2/x) dx

ln(Ky) = 2.ln(x) = ln(x^2)

Ky = x^2

y = Bx^2

The orthogonal family of curves is: y = Bx^2