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 $PQ.$
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1Does this answer your question? Geometrical proof for length of chord passing through vertex of parabola – Clemens Bartholdy Aug 05 '21 at 08:47
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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.
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Proof of first statement:https://math.stackexchange.com/questions/475666/prove-that-in-a-parabola-the-tangent-at-one-end-of-a-focal-chord-is-parallel-to?rq=1 – Clemens Bartholdy Jul 14 '21 at 06:17