Form the differential equation representing the family of curves given by $(x - a)^{2} + 2 y^{2} = a^{2},$ where a is an arbitrary constant.

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Solution

Given, family of curves is $(x - a)^{2} + 2 y^{2} = a^{2}, a$ being an arbitrary constant.
$\Rightarrow x^{2} - 2 a x + 2 y^{2} = 0$ ...(i)
On differentiating both sides w.r.t. x, we get
$2 x - 2 a + 4 y \frac{d y}{d x} = 0$ ...(ii)
On multiplying Eq. (ii) by x and substracting Eq. (i) from it, we get
$x (2 x - 2 a + 4 y \frac{d y}{d x}) - (x^{2} - 2 a x + 2 y^{2}) = 0 \Rightarrow 2 x^{2} - 2 a x + 4 x y \frac{d y}{d x} - x^{2} + 2 a x - 2 y^{2} = 0 \Rightarrow 4 x y \frac{d y}{d x} + x^{2} - 2 y^{2} = 0 \Rightarrow \frac{d y}{d x} = \frac{2 y^{2} - x^{2}}{4 x y}$
which is the required differential equation.

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