Parabola | Problem 7 | DETERMINE THE VERTEX, FOCUS, DIRECTRIX, AND LATUS RECTUM | Judd Hernandez |

Summary

This video, presented by Judd Hernandez, demonstrates how to determine the vertex, focus, directrix, and length of the latus rectum of a parabola given its equation. The process involves completing the square to transform the equation into a standard form, then extracting the necessary information.

Highlights

Introduction to Parabola Equations00:00:00

This video will focus on determining the vertex, focus, directrix, and latus rectum from a given parabola equation. The first step in solving such equations is to complete the square.

Identifying Parabola Orientation based on the Squared Term00:01:09

If the equation has a y-squared term, the parabola opens either to the left or to the right. If it has an x-squared term, it opens either upward or downward. The specific direction of opening (left/right or up/down) is determined by the sign of the constant when the equation is in standard form.

Completing the Square and Standard Form00:01:54

The video demonstrates completing the square for the equation \(y^2 - 5x + 12y = -16\). After rearrangement and completing the square for the y-terms, the equation becomes \((y+6)^2 = 5(x+4)\). This matches the standard form \((y-k)^2 = 4c(x-h)\).

Determining the Vertex and Orientation00:03:20

From the standard form, the vertex (h, k) is identified as (-4, -6). Since the y-term is squared and the right side of the equation (5) is positive, the parabola opens to the right.

Calculating 'c' and the Latus Rectum00:04:13

The term 4c in the standard form is equal to 5. Therefore, \(c = 5/4 = 1.25\). The length of the latus rectum (LR) is \(|4c|\), which in this case is \(5\) units.

Finding the Focus00:04:56

Since the parabola opens to the right, the focus will be to the right of the vertex. The y-coordinate of the focus will be the same as the vertex (-6). The x-coordinate is found by adding 'c' to the x-coordinate of the vertex: \(-4 + 1.25 = -2.75\). So, the focus is at \((-2.75, -6)\) or \((-11/4, -6))\).

Finding the Directrix00:06:29

The directrix is a vertical line for parabolas opening horizontally, located 'c' units from the vertex in the opposite direction of the focus. So, \(x = -4 - 1.25 = -5.25\). The directrix is \(x = -5.25\) or \(x = -21/4)\).

Summary of Results00:07:44

By completing the square, the standard equation of the parabola was found to be \((y + 6)^2 = 5(x + 4))\. From this, the vertex is \((-4, -6))\), the focus is \((-2.75, -6))\), the directrix is \(x = -5.25))\), and the length of the latus rectum is \(5\) units.