Question

Let $A,B$ 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 diameter, the slope of the line $AB$ is

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

Answer 1

Answer: (C), (D)

Centre of circle $=\left(\frac{t_1^2+t_2^2}{2},(t_1+t_2)\right)$

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 $AB=\frac{2}{t_1+t_2}=\frac{2}{\pm r}$