Let A and B be two distinct points on the par | vedclass

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(C) Let the coordinates of points $A$ and $B$ be $(t_1^2, 2t_1)$ and $(t_2^2, 2t_2)$ respectively.The center of the circle with diameter $AB$ is the m

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$(C, D)$

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(C) Let the coordinates of points $A$ and $B$ be $(t_1^2, 2t_1)$ and $(t_2^2, 2t_2)$ respectively.The center of the circle with diameter $AB$ is the midpoint of $AB$,which is $\left(\frac{t_1^2+t_2^2}{2}, t_1+t_2\right)$.The radius of the circle is $r$. Since the axis of the parabola (the $x$-axis,$y=0$) touches the circle,the absolute value of the $y$-coordinate of the center must be equal to the radius $r$.Thus,$|t_1+t_2| = r$,which implies $t_1+t_2 = \pm r$.The slope $m$ of the line joining $A$ and $B$ is given by $m = \frac{2t_2 - 2t_1}{t_2^2 - t_1^2} = \frac{2(t_2 - t_1)}{(t_2 - t_1)(t_2 + t_1)} = \frac{2}{t_1+t_2}$.Substituting $t_1+t_2 = \pm r$,we get $m = \pm \frac{2}{r}$.Therefore,the possible values for the slope are $\frac{2}{r}$ and $-\frac{2}{r}$,which correspond to options $C$ and $D$.

Let $A$ and $B$ be two distinct points on the parabola $y^2 = 4x$. If the axis of the parabola touches a circle of radius $r$ having $AB$ as its diameter,then the slope of the line joining $A$ and $B$ can be

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