Question

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 $AB$ can be

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

Answer 1

Answer: (C), (D)

The correct options are
C $-\frac{2}{r}$
D $\frac{2}{r}$

$A=(t_1^2,{2t}_1),B=(t_2^2,{2t}_2)$
Center $=[\frac{t_1^2+t_2^2}{2},(t_1+t_2)]$
$t_1+t_2=\pm r$
$\therefore$ Slope $=\frac{2(t_1-t_2)}{t_1^2-t_2^2}=\frac{2}{t_1+t_2}=\pm \frac{2}{r}$