Question

If chord PQ subtends an angle $\theta $ at the vertex of ${{y}^{2}}=4ax$, then $\tan \theta =$

A. $\dfrac{2}{3}\sqrt{7}$
B. $\dfrac{-2}{3}\sqrt{7}$
C. $\dfrac{2}{3}\sqrt{5}$
D. $-\dfrac{2}{3}\sqrt{5}$

Answer 1

Hint: For solving this type of question we should know about the concept of chord or focal chord. First, we have to find the shape of the diagram, that is, if it is a circle, ellipse or a parabola and we can find this by the help of an equation which will be given in the question and then we can calculate the value of $\tan \theta $.

Complete step-by-step answer:
So, it is given in the question that the equation of the diagram is ${{y}^{2}}=4ax$ and this is the equation of the parabola. And the equation of the line is $y=2x+a$. So, for calculation of the angle made by the chord PQ at vertex (0, 0) is given by,
So, we can write,
$\begin{align}
  & \tan \theta =\dfrac{\left( \dfrac{2}{t}+2t \right)}{1-4} \\
 & \Rightarrow \tan \theta =\dfrac{2\left( \dfrac{1}{t}+t \right)}{-3} \\
 & \Rightarrow \tan \theta =-\dfrac{2}{3}\left( \dfrac{1}{t}+t \right) \\
\end{align}$
Since, ${{\left( \dfrac{1}{t}+t \right)}^{2}}=5$
So, $\tan \theta =\dfrac{-2}{3}\sqrt{5}$
Here $\tan \theta $ is calculated by the formula of $\tan \theta $ which is equal to the opposite/adjacent.
If any line $y=mx+c$ is a tangent to the parabola, then the equation for the figure is ${{y}^{2}}=4ax$ and this is valid if $c=\dfrac{a}{m}$.

Note: In this type of question, it can also be asked to find the point of contact, in that case you should know about them too. Generally, the point of contact of the tangent $y{{y}_{1}}=2a\left( x+{{x}_{1}} \right)$ and with the parabola ${{y}^{2}}=4ax$ is given by $\left( {{x}_{1}},{{y}_{1}} \right)$ point form, and the point of contact of the tangent is given as $y=mx+\dfrac{a}{m}$.