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

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(D) 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 $

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(D) 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 given by $(\frac{t_{1}^{2}+t_{2}^{2}}{2}, t_{1}+t_{2})$.The axis of the parabola $y^{2}=4x$ is the $x$-axis,which has the equation $y=0$.Since the circle touches the $x$-axis,the radius of the circle is equal to the absolute value of the $y$-coordinate of the center.Thus,$r = |t_{1}+t_{2}|$,which implies $t_{1}+t_{2} = \pm r$.The slope of the line $AB$ 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 the slope $m = \frac{2}{\pm r} = \pm \frac{2}{r}$.Since $\pm \frac{2}{r}$ is not explicitly listed in the options $A, B,$ or $C$,the correct choice is $D$.

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 $AB$ is

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