Question

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

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

Answer 1

The correct option is D

$2$

Explanation for the correct option:

Finding the value of $\tan \theta$:

Given parabola, $y^2=16x$.

Then, $\frac{dy}{dx}=\frac{16}{2y}=\frac{8}{y}$.

So, the slope of the tangent to the given parabola at $P(16,16)$$={\left(\frac{dy}{dx}\right)}_{(16,16)}={\left(\frac{8}{y}\right)}_{(16,16)}=\frac{8}{16}=\frac{1}{2}.$

and the slope of the normal to the given parabola at

$P(16,16)={\left(-\frac{dx}{dy}\right)}_{(16,16)}={\left(-\frac{y}{8}\right)}_{(16,16)}=-\frac{16}{8}=-2$.

Then the equation of the tangent at $P(16,16)$ is

$$\begin{gathered} y-16=\frac{1}{2}\left(x-16\right) \\ \Rightarrow 2y-32=x-16 \\ \Rightarrow x-2y+16=0 \end{gathered}$$

and the equation of the normal at $P(16,16)$ is

$$\begin{gathered} y-16=-2\left(x-16\right) \\ \Rightarrow 2x+y-48=0 \end{gathered}$$

The tangent and normal intersect the axis of the parabola at $A(-16,0)\text{and}B(24,0)$respectively

Since, $C$ is the center of the circle through the points $A,P\text{and}B$ and $AP\perp PB$, So, $AB$ is the diameter of the circle and hence $C$ is the mid-point of the line segment $AB$ and hence the coordinates of $C$ would be $\left(\frac{-16+24}{2},0\right)=\left(4,0\right).$

Now, the slope of the line segment $PC=m_1=\frac{16-0}{16-4}=\frac{16}{12}=\frac{4}{3}.$

Again, the slope of the line segment $PB=m_2=\frac{16-0}{16-24}=\frac{16}{-8}=-2.$

Hence

$$\begin{gathered} \tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right| \\ =\left|\frac{{\displaystyle \frac{4}{3}}-\left(-2\right)}{1+{\displaystyle \frac{4}{3}}\times \left(-2\right)}\right| \\ =\left|\frac{{\displaystyle \frac{10}{3}}}{{\displaystyle \frac{-5}{3}}}\right| \\ =\left|\frac{10}{-5}\right| \\ =2 \end{gathered}$$

Therefore, the correct answer is option (D).