Parabola 2 Question 6
6. Tangent and normal are drawn at $P(16,16)$ on the parabola $y^{2}=16 x$, 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 $\angle C P B=\theta$, then a value of $\tan \theta$ is
(2018 Main)
(a) $\frac{1}{2}$
(b) 2
(c) 3
(d) $\frac{4}{3}$
Show Answer
Answer:
Correct Answer: 6. (c)
Solution:
- Equation of tangent and normal to the curve $y^{2}=16 x$ at $(16,16)$ is $x-2 y+16=0$ and $2 x+y-48=0$, respectively.
$$ A=(-16,0) ; \quad B=(24,0) $$
$\because C$ is the centre of circle passing through $P A B$
i.e.
$$ C=(4,0) $$
Slope of $P C=\frac{16-0}{16-4}=\frac{16}{12}=\frac{4}{3}=m _1$
Slope of $P B=\frac{16-0}{16-24}=\frac{16}{-8}=-2=m _2$
$$ \tan \theta=\frac{m _1-m _2}{1+m _1 m _2} $$
$$ \Rightarrow \quad \tan \theta=\frac{\frac{4}{3}+2}{1-\frac{4}{3}(2)} \Rightarrow \tan \theta=2 $$
