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)

$\frac{2}{r}$
$-\frac{2}{r}$

Let $A(t_2^2,2t_2),B(t_1^2,2t_1)$ be the two points on the parabola.

$AB$ is the diameter of the circle.

Let $C$ be the centre of the circle.

$⟹C\equiv (\frac{t_1^2+t_2^2}{2},t_1+t_2)$

The axis of the parabola is $X-axis$

So, the circle touches the $X-axis$

Hence, distance of $C$ from $X-axis=$ radius of circle

$⟹|t_1+t_2|=r$

$⟹(t_1+t_2)=\pm r$

Slope of $AB=\frac{y_2-y_1}{x_2-x_1}$

$=\frac{2t_1-2t_2}{t_1^2-t_2^2}$

$=\frac{2}{t_1+t_2}$

$=\pm \frac{2}{r}$

Hence, the answer is an option (C) and (D)