Question
Which of the following does not represent the orthogonal trajectory of the system of curves $${\left( {\frac{{dy}}{{dx}}} \right)^2} = \frac{a}{x}$$
A.
$$9a{\left( {y + c} \right)^2} = 4{x^3}$$
B.
$$y + c = \frac{{ - 2}}{{3\sqrt a }}{x^{\frac{3}{2}}}$$
C.
$$y + c = \frac{2}{{3\sqrt a }}{x^{\frac{3}{2}}}$$
D.
All are orthogonal trajectories
Answer :
All are orthogonal trajectories
Solution :
The family of curves which are orthogonal (i.e. intersect at right angles) to a given system of curves is obtained by substituting $$ - \frac{{dx}}{{dy}}$$ for $$\frac{{dy}}{{dx}}$$ in the differential equation of the given system.
The given differential equation is $${\left( {\frac{{dy}}{{dx}}} \right)^2} = \frac{a}{x}$$
Replacing $$\frac{{dy}}{{dx}}$$ by $$ - \frac{{dx}}{{dy}},$$ we get
$${\left( {\frac{{dx}}{{dy}}} \right)^2} = \frac{a}{x}\, \Rightarrow {\left( {\frac{{dy}}{{dx}}} \right)^2} = \frac{x}{a}\, \Rightarrow \frac{{dy}}{{dx}} = \pm \sqrt {\frac{x}{a}} $$
Integrating we get
$$\eqalign{
& y + c = \pm \frac{2}{{3\sqrt a }}{x^{\frac{3}{2}}}......\left( {\text{i}} \right) \cr
& \Rightarrow {\left( {y + c} \right)^2} = \frac{4}{{9a}}{x^3} \cr
& \Rightarrow 9a{\left( {y + c} \right)^2} = 4{x^3}......\left( {{\text{ii}}} \right) \cr} $$
From $$\left( {{\text{i}}} \right)$$ and $$\left( {{\text{ii}}} \right)$$ all of the first three given options represent required equations.