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 $A B=\frac{2}{t_{1}+t_{2}}=\frac{2}{\pm r}$