Let chord PQ of the parabola y^2=12x have length 3√13, such that the ordinates of points P…

Question

Let chord $PQ$ of the parabola $y^2=12x$ have length $3\sqrt{13}$, such that the ordinates of points $P$ and $Q$ are in the ratio $1:2$. If the chord $PQ$ subtends an angle $\alpha$ at the focus of the parabola, then $\sin\alpha$ is equal to:

Accepted Answer

Here $a=3$; take $P=(3t^2,6t),\ Q=(12t^2,12t)$ (ordinate ratio $1:2$).

$$PQ^2=(9t^2)^2+(6t)^2=81t^4+36t^2=117 \;\Rightarrow\; 9t^4+4t^2-13=0 \;\Rightarrow\; (t^2-1)(9t^2+13)=0\;\Rightarrow\;t=1.$$

So $P=(3,6),\ Q=(12,12)$, focus $S=(3,0)$. $PS$ is vertical; slope of $QS=\dfrac{12}{9}=\dfrac43$, so the angle between $QS$ and the vertical has $\tan\alpha=\dfrac34\Rightarrow\sin\alpha=\dfrac35$.

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