Question

Show that the locus of the midpoints of chords of the parabola y² = 4ax which are of constant length 2l is (4ax - y²)(y² + 4a²) = 4a²l².

Answer 1

Let P(at₁²,2at₁) and Q(at₂², 2at₂)be the extremities of a chord of length 2l and M(x₁, y₁) be the midpoint of PQ so that

By hypothesis, the length of the segment PQ is

Now, from Eqs. (1) and (2), we get

which imply that

Substituting the value of t₁t₂ from Eq. (2) and t₁ + t₂ = y₁a in Eq. (1), We have

Hence, the locus of (x₁, t₁) is

(4ax - y²)(y² + 4a²) = 4a²l²