length of the focal chord

Question

Paragraph:

$PQ$ is a focal chord of the parabola: $y^2=4ax.$

The tangents to the parabola at the points $P$ and $Q$ meet at point $R$ which lies on the line $y=2x + a.$

Question:

Find the length of the chord $PQ.$

Answer 1

Find the point R. It comes out to be (-a,-a)

And then we take the points P and Q to be represented parametrically by c and d.

Thus we have:

c + d = -1 ...(1)

If we take the points P and Q to be the parametric points c and d on the parabola then we have the length of the chord to be:

Length = PS + QS

= a + a + ac² + ad²

= 2a + a(c²+d²)

We need the value of (c² + d²)

Since:

cd = -1

c + d = -1 (from (1))

(c + d)² = c² + d² + 2cd

1 = c² + d² -2

c² + d² = 3

=> Thus length = 5a.