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

• A

  $-\frac{1}{r}$
• B

  $\frac{1}{r}$
• Correct answer
  C

  $\frac{2}{r}$
• D

  None of the above

Solution

Centre of circle $=\left(\frac{t_{1}^{2}+t_{2}^{2}}{2},\left(t_{1}+t_{2}\right)\right)$ 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 $A B=\frac{2}{t_{1}+t_{2}}=\frac{2}{\pm r}$

Hence, the correct answer is option: 'C'.