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(C) Statement $I$: The equation of a circle with center $(0, \alpha)$ and radius $k$ is $x^2 + (y - \alpha)^2 = k^2$ ... $(i)$.Differentiating with re

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Both Statement $I$ and Statement $II$ are true

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(C) Statement $I$: The equation of a circle with center $(0, \alpha)$ and radius $k$ is $x^2 + (y - \alpha)^2 = k^2$ ... $(i)$.Differentiating with respect to $x$: $2x + 2(y - \alpha)\frac{dy}{dx} = 0 \Rightarrow y - \alpha = -x\frac{dx}{dy}$.Thus, $\alpha = y + x\frac{dx}{dy}$.Substituting $\alpha$ into $(i)$: $x^2 + (-x\frac{dx}{dy})^2 = k^2 \Rightarrow x^2 + x^2(\frac{dx}{dy})^2 = k^2$.$x^2(1 + (\frac{dx}{dy})^2) = k^2 \Rightarrow x^2(1 + \frac{1}{(dy/dx)^2}) = k^2 \Rightarrow x^2(\frac{(dy/dx)^2 + 1}{(dy/dx)^2}) = k^2$.$x^2(dy/dx)^2 + x^2 = k^2(dy/dx)^2 \Rightarrow (x^2 - k^2)(dy/dx)^2 + x^2 = 0$. Hence, Statement $I$ is true.Statement $II$: The circle passes through the origin and its center is on the $X$-axis, so center is $(\alpha, 0)$ and radius is $|\alpha|$.The equation is $(x - \alpha)^2 + y^2 = \alpha^2 \Rightarrow x^2 - 2x\alpha + \alpha^2 + y^2 = \alpha^2 \Rightarrow x^2 + y^2 = 2x\alpha \Rightarrow \alpha = \frac{x^2 + y^2}{2x}$.Differentiating $x^2 + y^2 = 2x\alpha$ with respect to $x$: $2x + 2y\frac{dy}{dx} = 2\alpha$.Substitute $\alpha$: $x + y\frac{dy}{dx} = \frac{x^2 + y^2}{2x} \Rightarrow 2x^2 + 2xy\frac{dy}{dx} = x^2 + y^2 \Rightarrow x^2 - y^2 + 2xy\frac{dy}{dx} = 0$. Hence, Statement $II$ is true.

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