Form the differential equation by eliminating the arbitrary constant from the equation $x^{2} + y^{2} - 2 a y - a^{2} = 0$

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$(2 y^{2} + x^{2}) {(\frac{d y}{d x})}^{2} + 4 x y \frac{d y}{d x} + x^{2} = 0$

$(2 y^{2} - x^{2}) {(\frac{d y}{d x})}^{2} + 4 x y \frac{d y}{d x} + x^{2} = 0$

$(2 y^{2} - x^{2}) {(\frac{d y}{d x})}^{2} - 4 x y \frac{d y}{d x} + x^{2} = 0$

$(2 y^{2} - x^{2}) {(\frac{d y}{d x})}^{2} - 4 x y \frac{d y}{d x} - x^{2} = 0$

answer is B.

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Detailed Solution

$\begin{array}{l} x^{2} + y^{2} - 2 a y - a^{2} = 0 \Rightarrow 2 x + 2 y y_{1} - 2 a y_{1} = 0 \Rightarrow a = \frac{x + y y_{1}}{y_{1}} = \frac{x}{y_{1}} + y \\ ∴ x^{2} + y^{2} - 2 y (\frac{x}{y_{1}} + y) - {(\frac{x}{y_{1}} + y)}^{2} = 0 \Rightarrow (x^{2} + y^{2}) y_{1}^{2} - 2 y_{1} y (x + y y_{1}) - {(x + y y_{1})}^{2} = 0 \\ \Rightarrow (x^{2} - 2 y^{2}) y_{1}^{2} - 2 x y y_{1} - x^{2} - 2 x y y_{1} = 0 \Rightarrow (x^{2} - 2 y^{2}) y_{1}^{2} - 4 x y y_{1} - x^{2} = 0 \\ \Rightarrow (2 y^{2} - x^{2}) {(\frac{d y}{d x})}^{2} + 4 x y \frac{d y}{d x} + x^{2} = 0 . \end{array}$