The locus of the middle points of chords of the parabola y …

Maths Conic Sections Parabola Single Correct MCQ

Published by: CGP EDU Academic Team

Published on: August 13, 2026

The locus of the middle points of chords of the parabola y 2 = 4ax, which are of constant length 2 λ is-

(4x + y 2 ) (y 2 – 4) = 4 λ 2

(4y + x 2 ) (x 2 – 4) = 4 λ 2

(4x – y 2 ) (y 2 + 4) = 4 λ 2

None of these

Text Solution

Verified by Experts

The correct answer is:

Let R ≡ (h, k) be mid-point on locus

P ≡ (x 1 , y 1 ) and Q ≡ (x 2 , y 2 ) be points on parabola

forming a chord so 2h = x 1 + x 2 , 2k = y 1 + y 2

we have (x 1 – x 2 ) 2 + (y 1 – y 2 ) 2 = (2 λ ) 2 also P, Q lies on parabola ⇒ y 1

2 = 4x 1 , y 2 2 = 4x 2

so + (y 1 – y 2 ) 2 = 4 λ 2 ⇒ (y 1 – y 2 )

2 (k 2 + 4) = 16 λ 2 2y 1 y 2 = 4k

2 – 8h ⇒ (y 1 – y 2 ) 2 = 16h – 4k 2 So we have (16h – 4k

2 ) (k 2 + 4) = 16 λ 2 So locus (4x – y

2 ) (y 2 + 4) = 4 λ 2

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