In coordinate geometry, what is meant by the latus rectum of a parabola?

Introduction / Context:
The term latus rectum appears frequently in coordinate geometry, especially in questions dealing with standard parabolic equations, focal properties, and lengths related to conic sections. Understanding exactly what geometric object the latus rectum represents is essential before applying formulas for its length in problems involving parabolas.

Given Data / Assumptions:

• The conic under discussion is specifically a parabola.
• The options mention line segment, vertex, focus, major axis, and directrix.
• We assume the standard geometric definition used in mathematics textbooks.
• The latus rectum is related to the focus and the directrix of the parabola.

Concept / Approach:
A parabola can be defined as the set of points equidistant from a fixed point called the focus and a fixed line called the directrix. The latus rectum is a special chord (line segment) of the parabola. It passes through the focus and is parallel to the directrix. Its endpoints lie on the parabola, and its length is often used in standard formulas, for example 4a for the parabola y^2 = 4ax.

Step-by-Step Solution:
Step 1: Recall that a parabola has a focus (a fixed point) and a directrix (a fixed line). Step 2: The latus rectum is defined as a line segment whose endpoints lie on the parabola. Step 3: This line segment passes through the focus of the parabola. Step 4: The latus rectum is parallel to the directrix of the parabola. Step 5: Because of these properties, the latus rectum is a chord that helps describe the width of the parabola near its focus. Step 6: Therefore the best description is that it is a line segment through the focus and parallel to the directrix.

Verification / Alternative check:
In the standard parabola y^2 = 4ax, the focus is at (a, 0) and the directrix is x = -a. The latus rectum has endpoints at (a, 2a) and (a, -2a), which are points on the parabola. The segment joining these points has x coordinate a for all points and is therefore vertical, while the directrix is the vertical line x = -a. So the latus rectum is indeed parallel to the directrix and passes through the focus at (a, 0), matching the definition used in the correct option.

Why Other Options Are Wrong:

• The vertex of the parabola is a single point where the curve changes direction, not a line segment.
• The focus is also a single point, not the chord that passes through it.
• The major axis is a concept used mainly for ellipses, not for simple parabolas.
• The directrix is an infinite straight line, not the finite line segment inside the parabola.

Common Pitfalls:
Many learners confuse the focus and the latus rectum, treating the latus rectum as if it were a single point. Others mix up definitions related to ellipses and parabolas, especially the term major axis. Remember that in a parabola the important objects are the vertex, focus, directrix, and the chord through the focus that is parallel to the directrix, which is the latus rectum.

Final Answer:
The latus rectum of a parabola is a line segment through the focus and parallel to the directrix of the parabola.