The locus of the mid-point of all chords of the parabola y^2 = 4x which are d...

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Question Id : 11519 | Context :NIMCET 2023

The locus of the mid-point of all chords of the parabola

$y^2 = 4x$ which are drawn through its vertex is

🎥 Video solution / Text Solution of this question is given below:

Given Parabola: $y^2 = 4x$

Condition: Chords pass through the vertex $(0, 0)$

Let the other end of the chord be $(x_1, y_1)$ , so the midpoint is:

$M = \left( \frac{x_1}{2}, \frac{y_1}{2} \right) = (h, k)$

Since the point lies on the parabola: $y_1^2 = 4x_1$

⇒ $(2k)^2 = 4(2h)$

⇒ $4k^2 = 8h$

⇒ $\boxed{k^2 = 2h}$

✅ Locus of midpoints: $y^2 = 2x$

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