Consider the following statements:I. The inte | vedclass

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(C) Statement $I$: For the circle $x^2+y^2-2x-4y+1=0$,the intercept on the $Y$-axis is given by $2\sqrt{f^2-c}$,where $f = -2$ and $c = 1$. Thus,the i

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(C) Statement $I$: For the circle $x^2+y^2-2x-4y+1=0$,the intercept on the $Y$-axis is given by $2\sqrt{f^2-c}$,where $f = -2$ and $c = 1$. Thus,the intercept is $2\sqrt{(-2)^2-1} = 2\sqrt{4-1} = 2\sqrt{3}$. So,statement $I$ is True.Statement $II$: For the circle $x^2+y^2-4x-2y+6=0$,the intercept on the $X$-axis is given by $2\sqrt{g^2-c}$,where $g = -2$ and $c = 6$. Thus,the intercept is $2\sqrt{(-2)^2-6} = 2\sqrt{4-6} = 2\sqrt{-2}$. Since the value inside the square root is negative,the circle does not intersect the $X$-axis. So,statement $II$ is False.Statement $III$: The line $y=2x+1$ or $2x-y+1=0$ intersects the circle $x^2+y^2=9$ at two distinct points if the perpendicular distance $p$ from the center $(0,0)$ to the line is less than the radius $r=3$. Here,$p = \frac{|2(0)-(0)+1|}{\sqrt{2^2+(-1)^2}} = \frac{1}{\sqrt{5}}$. Since $p = \frac{1}{\sqrt{5}} Therefore,$I$ is True,$II$ is False,and $III$ is True. The correct option is $(c)$.

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