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If c is a parameter, then differential equation of the family of curves $x^{2} = c {(y + c)}^{2}$ is

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$x {(\frac{d y}{d x})}^{3} + y {(\frac{d y}{d x})}^{2} - 1 = 0$

$x {(\frac{d y}{d x})}^{3} - y {(\frac{d y}{d x})}^{2} + 1 = 0$

$x {(\frac{d y}{d x})}^{3} + y {(\frac{d y}{d x})}^{2} + 1 = 0$

$x {(\frac{d y}{d x})}^{3} - y {(\frac{d y}{d x})}^{2} - 1 = 0$

answer is D.

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Detailed Solution

$x^{2} = c {(y + c)}^{2}$

$2 x = c .2. (y + c) . \frac{d y}{d x} S$

$x = c (y + c) . (\frac{d y}{d x})$ ….(1)

$x^{2} = c^{2} {(y + c)}^{2} {(\frac{d y}{d x})}^{2}$

$\frac{x^{2}}{x^{2}} = \frac{c {(y + c)}^{2}}{c^{2} {(y + c)}^{2} {(\frac{d y}{d x})}^{2}}$

$c = \frac{1}{{(\frac{d y}{d x})}^{2}}$

Sub “c” in …………(1)

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