Question

Tangent and normal are drawn at $P(16,16)$ on the parabola $y^2=16x$, which intersect the axis of the parabola at $A$ and $B$, respectively. If $C$ is the centre of the circle through the points $P,A$ and $B$ and $\angle CPB=\theta$, then a value of $\tan theta$ is:

• A. $\frac{1}{2}$
• B. 2
• C. 3
• D. $\frac{4}{3}$

Answer 1

Answer: (B)

$y^2=16x$
Tangent at $P(16,16)$ is $2y=x+16...(1)$
Normal at $P(16,16)$ is $y=-2x+48...(2)$
i.e., $A$ is $(-16,0);B$ is $(24,0)$
Now, Centre of circle is $(4,0)$
Now, $m_{PC}=\frac{4}{3}$
$m_{PB}=-2$
i.e., $\tan \theta =\left|\frac{\frac{4}{3}+2}{1-\frac{8}{3}}\right|=2$