Question

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

• A. $-\frac{1}{r}$
• B. $\frac{1}{ r }$
• C. $\frac{2}{ r }$
• D. $-\frac{2}{r}$

Answer 1

Answer: (C), (D)

$A =\left( t _{1}^{2}, 2 t _{1}\right), B =\left( t _{2}^{2}, 2 t _{2}\right)$
Centre $=\left[\frac{ t _{1}^{2}+ t _{2}^{2}}{2},\left( t _{1}+ t _{2}\right)\right]$
$t _{1}+ t _{2}=\pm r$
$m =\frac{2\left( t _{1}- t _{2}\right)}{ t _{1}^{2}- t _{2}^{2}}=\frac{2}{ t _{1}+ t _{2}}=\pm \frac{2}{ r }$