D.E of the parabolas having x-axis as axis and origin as focus is:

y {(\frac{d y}{d x})}^{2} + 4 x \frac{d y}{d x} = 4

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2 x \frac{d y}{d x} - y = 0

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

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

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Solution

The correct option is D $y {(\frac{d y}{d x})}^{2} + 2 x \frac{d y}{d x} - y = 0$
$\Rightarrow$ The equation of parabola became
$\sqrt{(x + 2 a)^{2}} = \sqrt{x^{2} + y^{2}}$
$\Rightarrow (x + 2 a)^{2} = x^{2} + y^{2}$
$\Rightarrow 4 a x + 4 a^{2} = y^{2}$
$\Rightarrow 2 y y_{1} = 4 a$
$\Rightarrow 2 a = y y_{1}$
$\Rightarrow y^{2} = 2 y y_{1} x + y^{2} y_{1}^{2}$
$\Rightarrow y = 2 y_{1} x + y y_{1}^{2}$
$\Rightarrow y {(\frac{d y}{d x})}^{2} + 2 x (\frac{d y}{d x}) - y = 0$

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D.E of the parabolas having x-axis as axis and origin as focus is:

The correct option is D y(dydx)2+2xdydx−y=0⇒ The equation of parabola became√(x+2a)2=√x2+y2⇒(x+2a)2=x2+y2⇒4ax+4a2=y2⇒2yy1=4a⇒2a=yy1⇒y2=2yy1x+y2y21⇒y=2y1x+yy21⇒y(dydx)2+2x(dydx)−y=0