How To Find Latus Rectum Of Parabola

How to Find the Latus Rectum of a Parabola: A Complete Guide

The latus rectum is one of the most elegant and useful geometric properties of a parabola. It is a special chord—a line segment with both endpoints on the curve—that passes through the focus and is perpendicular to the axis of symmetry. Its length is a direct measure of how "wide" or "narrow" the parabola opens. Understanding how to find the latus rectum, both its length and its precise endpoints, is a fundamental skill in conic sections, with applications in physics (satellite dishes, projectile paths), engineering (reflector design), and advanced mathematics. This guide will break down the process for any parabola, regardless of its orientation or position.

Understanding the Core Concept: What is the Latus Rectum?

Before calculating, we must be perfectly clear on the definition. For a parabola, the latus rectum is the focal chord that is perpendicular to the axis of symmetry. Its most important feature is its length, which is universally designated as 4p, where p is the distance from the vertex to the focus (the focal length).

• For a parabola that opens upward or downward (vertical axis), the latus rectum is a horizontal line segment.
• For a parabola that opens left or right (horizontal axis), the latus rectum is a vertical line segment.

The endpoints of the latus rectum are always at the same "height" (for vertical parabolas) or "horizontal position" (for horizontal parabolas) as the focus. This relationship is the key to finding both its length and its coordinates.

The Essential First Step: Put the Equation in Standard Form

You cannot find the latus rectum from a general quadratic equation like Ax² + Bxy + Cy² + Dx + Ey + F = 0 directly. The universal method requires the equation to be in one of its standard forms. This almost always involves completing the square.

There are four primary standard forms, grouped by orientation:

1. Vertical Parabola (opens up/down):

   • Vertex at (h, k): (x - h)² = 4p(y - k)
   • If p > 0, it opens upward. If p < 0, it opens downward.

2. Horizontal Parabola (opens left/right):

   • Vertex at (h, k): (y - k)² = 4p(x - h)
   • If p > 0, it opens right. If p < 0, it opens left.

Crucial Insight: In both forms, the coefficient of the non-squared term is 4p. Therefore, finding 4p is finding the length of the latus rectum. Your primary task is to manipulate the given equation until it matches one of these forms and read off the value of 4p.

Step-by-Step Procedures for Finding the Latus Rectum

Case 1: Equation is Already in Standard Form

This is the simplest scenario. Identify the form and the value of 4p.

• Example (Vertical): (x - 3)² = 8(y + 1)

  • Compare to (x - h)² = 4p(y - k). Here, 4p = 8.
  • Length of Latus Rectum = 8 units.
  • The vertex is (3, -1). The focus is (h, k + p). Since 4p=8, p=2. So focus is (3, -1+2) = (3, 1).
  • The latus rectum is horizontal and passes through y=1. Its endpoints have x-coordinates h ± 2p (because from the focus, you move 2p left and right to reach the curve).
  • 2p = 4. So endpoints are (3-4, 1) = (-1, 1) and (3+4, 1) = (7, 1).

• Example (Horizontal): (y + 4)² = -12(x - 2)

  • Compare to (y - k)² = 4p(x - h). Here, 4p = -12.
  • Length of Latus Rectum = |4p| = 12 units. (Length is always positive).
  • Vertex is (2, -4). p = -3. Focus is (h + p, k) = (2-3, -4) = (-1, -4).
  • The latus rectum is vertical and passes through x = -1. Its endpoints have y-coordinates k ± 2p.
  • 2p = -6. So endpoints are (-1, -4-6) = (-1, -10) and (-1, -4+6) = (-1, 2).

Case 2: Equation is in General Quadratic Form (Requires Completing the Square)

This is the most common real-world problem. The strategy is to isolate the squared term and complete the square on the other variable.

General Algorithm:

1. Group terms by the squared variable and the linear term of the other variable.
2. Move the constant and the non-squared linear term to the other side.
3. Complete the square on the grouped terms.
4. Factor the perfect square trinomial.
5. Isolate the squared term to match the standard form (variable)² = 4p(other variable).
6. **Read `