A double ordinate of the parabola y^2=4 a x is of length 8...

A double ordinate of the parabola $y^{2}=4 a x$ is of length 8 a. Then the angle between the lines from the vertex to its ends is
(a) $\frac{\pi}{3}$
(b) $\frac{\pi}{2}$
(c) $\frac{\pi}{6}$
(d) $\pi$

Key Concepts

Angle Between Lines from a Common Point

Determining the angle between lines drawn from a common point, such as the vertex, to other points on the curve involves concepts from trigonometry and geometry. This angle can be calculated using vector methods or trigonometric ratios, and it provides insight into the spatial configuration of the curve relative to the vertex.

Parabola

A parabola is a type of conic section defined as the locus of all points equidistant from a given point (the focus) and a line (the directrix). Its standard equation reveals important geometric properties such as the vertex, focus, and axis of symmetry, which are crucial for setting up and solving related problems in coordinate geometry.

Double Ordinate

A double ordinate in a parabola is a chord that is perpendicular to the axis of symmetry. In the case of the parabola with equation y² = 4ax, the double ordinate is vertical. Understanding the double ordinate is important because its length, determined by the y-coordinates of the intersection points with the parabola, can be used to solve geometric problems involving angles and distances.

Chord Length in Conic Sections

The chord length between two points on a curve, such as on a parabola, is the distance measured by connecting the two points with a straight line. In parabolic problems, calculating the chord length, especially for specific chords like a double ordinate, involves substituting the coordinates from the parabola’s equation into the distance formula.

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