Parabola (Focus-Directrix Definition) Math Example 5

Follow the full solution, then compare it with the other examples linked below.

Example 5

hard
A parabola has vertex (2,−1)(2, -1) and focus (2,3)(2, 3). Find its equation and the length of the latus rectum.

Solution

  1. 1
    The focus is directly above the vertex, so the parabola opens upward. p=3−(−1)=4p = 3 - (-1) = 4. The equation is (x−2)2=4(4)(y+1)=16(y+1)(x-2)^2 = 4(4)(y+1) = 16(y+1).
  2. 2
    The latus rectum has length ∣4p∣=16|4p| = 16.

Answer

(x−2)2=16(y+1);latus rectum=16(x-2)^2 = 16(y+1); \quad \text{latus rectum} = 16
For a shifted parabola with vertex (h,k)(h,k), use (x−h)2=4p(y−k)(x-h)^2 = 4p(y-k). The latus rectum is the chord through the focus perpendicular to the axis, and its length is ∣4p∣|4p|. A larger latus rectum means a wider parabola.

About Parabola (Focus-Directrix Definition)

A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

Learn more about Parabola (Focus-Directrix Definition) →

More Parabola (Focus-Directrix Definition) Examples

Example 1 easy

Find the focus and directrix of the parabola [formula].

Example 2 medium

Write the equation of a parabola with vertex at the origin, opening to the right, with focus at [for

Example 3 medium

Find the focus and directrix of the parabola [formula].

Example 4 medium

Find the equation of the parabola with focus [formula] and directrix [formula].

← All Parabola (Focus-Directrix Definition) Examples Parabola (Focus-Directrix Definition) Concept