Question

Equation of a family of curve is $y^2=2kx+k^2$, what is the differential equation of the family of its orthogonal trajectory?

• A. ${\left(y{y}^{\prime }\right)}^2=-2xy{y}^{\prime }+y^2$
• B. ${\left(y{y}^{\prime }\right)}^2=-2xy{y}^{\prime }+x^2$
• C. ${\left(x{y}^{\prime }\right)}^2=-2x{y}^{\prime }-x^2$
• D. ${\left(x{y}^{\prime }\right)}^2=-2y{y}^{\prime }-y^2$

Answer 1

The correct option is A ${\left(y{y}^{\prime }\right)}^2=-2xy{y}^{\prime }+y^2$

We have, $y^2=2kx+k^2\left(1\right)$

Differentiating w.r.t. $x$, we get

$y{y}^{\prime }=k\left(2\right)$

Eliminating $k$, from (2) using (1), we get

$y^2=2xy{y}^{\prime }+{\left(y{y}^{\prime }\right)}^2$

For orthogonal trajectory, substituting $y^{\prime }$ with $-\frac{1}{y^{\prime }}$ in (2), we get

$y^2=2xy(-\frac{1}{y^{\prime }})+{\left(\frac{y}{y^{\prime }}\right)}^2$

$⟹{\left(y{y}^{\prime }\right)}^2=-2xy{y}^{\prime }+y^2$