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Let $A, B$ be two distinct points on the parabola $y^{2}=4 x$. If the axis of the parabola touches a circle of radius $r$ having $A B$ as diameter, the slope of the line $A B$ is
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The correct answer is:
$\frac{2}{r}$
Centre of circle $=\left(\frac{t_{1}^{2}+t_{2}^{2}}{2},\left(t_{1}+t_{2}\right)\right)$
since, circle touch the $x$ -axis, so equation of tangent is $y=0$
$\because$ Radius = Perpendicular distance from centre to the tangent
$\Rightarrow$ Radius $=\left|t_{1}+t_{2}\right|=r$
Slope of $A B=\frac{2}{t_{1}+t_{2}}=\frac{2}{\pm r}$
since, circle touch the $x$ -axis, so equation of tangent is $y=0$
$\because$ Radius = Perpendicular distance from centre to the tangent
$\Rightarrow$ Radius $=\left|t_{1}+t_{2}\right|=r$
Slope of $A B=\frac{2}{t_{1}+t_{2}}=\frac{2}{\pm r}$
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