A) \[9a{{(y+c)}^{2}}=4{{x}^{3}}\]
B) \[y+c=\frac{-2}{3\sqrt{a}}{{x}^{3/2}}\]
C) \[y+c=\frac{2}{3\sqrt{a}}{{x}^{3/2}}\]
D) All are orthogonal trajectories
Correct Answer: D
Solution :
[d] The family of curves which are orthogonal (i.e. intersect at right angles) to a given system of curves is obtained by substitute \[-\frac{dx}{dy}\] 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, \[y+c=\pm \frac{2}{3\sqrt{a}}{{x}^{3/2}}\] ? (i) \[\Rightarrow {{(y+c)}^{2}}=\frac{4}{9a}{{x}^{3}}\Rightarrow 9a{{(y+c)}^{2}}=4{{x}^{3}}\] .. (ii) From (i) and (ii) all of the first three given option represent required equations.
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