The differential equation for the family of curves $x^{2} - y^{2} + 2 a x = 0$ , where $a$ is an arbitrary constant is:

(x^{2} - y^{2}) = 2 x y y^{'}

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x^{2} - y^{2} = - 2 x y . y^{'}

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(x^{2} + y^{2}) y^{'} = 2 x y

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x^{2} + y^{2} = 2 x y . y^{'}

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Solution

The correct option is B $x^{2} + y^{2} = 2 x y . y^{'}$
$x^{2} - y^{2} + 2 a x = 0$ ......(i)
$\Rightarrow 2 x - 2 y y^{'} + 2 a = 0$
$\Rightarrow a = y y^{'} - x$
put value of a in equation (i)
$\Rightarrow x^{2} - y^{2} + 2 x y y^{'} - {2 x}^{2} = 0$
$\Rightarrow x^{2} + y^{2} = 2 x y . y^{'}$

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