Question

If a chord $4y = 3x - 48$ subtends an angle $\theta$ at the vertex of the parabola ${y^2} = 64x$ then $\tan \theta =$
A) $\dfrac{{10}}{9}$
B) $\dfrac{{13}}{9}$
C) $\dfrac{{20}}{9}$
D) $\dfrac{{16}}{9}$

Answer 1

In the given question, we have been given the equation of a chord. The chord subtends an angle at the vertex of the parabola whose equation has been given. We have to calculate the value of the given expression using such information and the given angle. To solve this, we are going to simplify the equation of the chord with the angle using the appropriate identity. Then we are going to deduce the normal values used in the equation of the parabola and substitute it into the formulae and calculate our answer.

Complete step by step answer:
The given equation of parabola is $y = 64x$.
Now, the normal equation of parabola is $Y = 4aX$
So, we have for this parabola, $a = 16$.
Hence, the focus is $\left( {16,0} \right)$.
Now, the end points of directrix is given by $P\left( {a{t^2},2at} \right)$ and $Q\left( {\dfrac{a}{{{t^2}}},\dfrac{{ - 2a}}{t}} \right)$.
Hence, for this parabola, we have,
$P\left( {16{t^2},32t} \right)$ and $Q\left( {\dfrac{{16}}{{2{t^2}}},\dfrac{{ - 2 \times 16}}{t}} \right) = Q\left( {\dfrac{8}{{{t^2}}},\dfrac{{ - 32}}{t}} \right)$
So, for the equation of the chord $4y = 3x - 48$, we have,
$4 \times 32t = 3 \times 16{t^2} - 48$
or $3{t^2} - 8t - 3 = 0$
Now, solving by splitting the middle term,
$3{t^2} - 9t + t - 3 = 0 \Rightarrow 3t\left( {t - 3} \right) + 1\left( {t - 3} \right) = 0$
Hence, $t = - \dfrac{1}{3},3$
Now, $\tan \theta = \tan \left( {{\theta _1} + {\theta _2}} \right)$
$= \dfrac{{\tan {\theta _1} + \tan {\theta _2}}}{{1 - \tan {\theta _1}\tan {\theta _2}}}$
So, for this parabola, we have,
$\dfrac{{\dfrac{{96}}{{144}} + \dfrac{{32/3}}{{16/9}}}}{{1 - \dfrac{{96}}{{144}} \times \dfrac{{32/3}}{{16/9}}}} = \dfrac{{96/144 + 6}}{{1 - 96/144 \times 6}}$
Thus, $\tan \theta = \dfrac{{20}}{9}$

Hence, the correct option is C.

Note:
In the given question, we were given the equation of a chord and the parabola. This chord subtended an angle at the vertex of the parabola. We had to calculate the value of the expression used in the question using such information and the given angle. We solved this by simplifying the equation of the chord with the angle using the appropriate identity. Then we deduced the normal values used in the equation of the parabola and substituted it into the formulae and calculated our answer. So, it is important that we know the formulae, identity and their results.