Parabola: If chord PQ subtends an angle theta at the vertex of y^2=4ax , then… | JEE Advanced 2013 Paper 2

Comprehension · Questions 1–1

Read the passage given below and answer the questions that follow.

Let $PQ$ be a focal chord of the parabola $y^2=4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a$, $a>0$.

JEE Advanced 2013 Paper 2Question 1

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

Parabola

$\frac23\sqrt7$$\frac{-2}{3}\sqrt7$$\frac23\sqrt5$$\frac{-2}{3}\sqrt5$✓ Correct answer

Solution

Take $a=1$: $x_1x_2=1$, $y_1y_2=-4$, $|y_1-y_2|=2\sqrt5$ (from $x_{1,2}=\frac{3\pm\sqrt5}{2}$). $\vec{OP}\cdot\vec{OQ}=1-4=-3<0$ so $\theta$ is obtuse, and slopes $m=4/y$ give $\tan\theta=\frac{m_1-m_2}{1+m_1m_2}=\frac{\pm2\sqrt5}{-3}$; with $\theta$ obtuse, $\tan\theta=-\frac23\sqrt5$.

Parabola - JEE Mathematics question (JEE Advanced 2013 Paper 2)

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