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  4. Tangent and normal are drawn at P(16, 16) on the parabola y2 = 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 = θ, then a value of tan theta is:

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Q. 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:

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Solution:

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