If (p, q) is the mid-point of a chord of the parabola y^2=4...

If $(p, q)$ is the mid-point of a chord of the parabola $y^{2}=4 x$, passing through its vertex then
(a) $\mathrm{p}^{2}=2 \mathrm{q}$
(b) $q^{2}=2 p$
(c) $\mathrm{p}^{2}=\mathrm{p}+\mathrm{q}$
(d) $q^{2}=p-q$

Key Concepts

Parabola

A parabola is a type of conic section defined by the property that any point on the curve is equidistant from a fixed point, the focus, and a fixed line, the directrix. The equation y² = 4x represents a standard horizontal parabola with its vertex at the origin, which has unique geometric properties including symmetry about its axis.

Chord of a Parabola

A chord in the context of a parabola is a line segment that connects two distinct points on the curve. Its properties are often explored to understand the geometric relationships and symmetries present in conic sections. When a chord passes through a specific point on the parabola, such as the vertex, it imposes additional constraints that can lead to special relationships between the coordinates of points along the chord.

Midpoint of a Chord

The midpoint of a chord in a parabola is obtained by averaging the coordinates of the endpoints of the chord. This midpoint holds significant geometric information, as its coordinates are often linked by simpler relations arising from the symmetrical nature of the parabola, especially when the chord has additional properties such as passing through the vertex.

Geometric Constraints from Vertex Passage

When a chord passes through the vertex of a parabola, it introduces a symmetry that simplifies the relationship between the chord’s endpoints and its midpoint. This extra condition reduces the degrees of freedom in the system, allowing for the derivation of specific algebraic relationships among the coordinates defined by the equation of the parabola.

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