Parabola as a Locus

The parabola is the main locus that is studied in Extension 1 Mathematics

You may be familiar with parabolas in the form of y = ax² + bx + c or y = (x-h)² +k

But in this topic we will be writing parabolas in a new form.

A parabola with the vertex at the origin will be written in the form of

x² = 4ay

The reason why we write parabolas in this new form is that we can easily find the vertex and focal length. This allows us to easily determine the point of the focus and the directrix. Just to recap a parabola is a locus of points that is equidistant from a fixed point called the focus and a line called the directrix.

Some definitions

Focal length: the length from the focus to the vertex, the length from the vertex to the directrix. It is equal to a.

Focus: the point a units from the vertex within the concave side of the parabola.

Directrix: the line a units from the vertex on the opposite side of the concave side.

Chord: a line that touches the curve twice.

Focal chord: a chord that goes through the focus.

Latus rectum: a focal chord that is perpendicular to the directrix. It has a length of 4a.

Diagram of parabola x² = 4ay

Four standard parabolas:

x² = 4ay

x² = -4ay

y² = 4ax

y² = -4ax

Vertex at point (h,k)

If the vertex of the parabola was at point (h,k) then we would replace x with (x-h) and y with (y-k) just like in transformation of graphs.

Example parabola question

1. Write down the equation of the parabola with vertex at (-1,2) and directrix at x=2

x=2 is vertical and the vertex is on the left side of the directrix.

This implies that the parabola will be in the form of (y-k)² = -4a(x-h)

(h,k) => (-1,2)

Therefore (y-2)² = -4a(x+1)

a = length from focus to vertex or length from directrix to vertex = 3units

so the equation of the parabola is (y-2)² = -12(x+1)

2. Find the focal length, focus, directrix and end points of the latus rectum of the parabola y= -3 -4x -x²

y = -(x² + 4x + 3)

   = -(x +4x + 4 -1)

   = -[(x+2)² - 1]

y - 1 = -(x+2)²

4a = 1

a = 1/4

Focal length is 1/4 units

Vertex at (-2 , 1)

The parabola is concave down

so the focus will be at (-2 , 3/4)

directrix is y = 5/4

Latus rectum is 4(1/4) = 1 units in length

Therefore the endpoints must be half unit on the left and right of the focus.

Endpoints are (-5/2 , 3/4) and (-3/2 , 3/4)