Question

A parabola has a focus of F(2, -0.5) and a directrix of y=-1.5 P(x,y) represents any point on the parabola, while D(x, -1.5) represents any point on the directrix.

What are the steps required to find the equation of the parabola?

Answer 1

The sketch of the parabola is attached below

We have the focus
`(a,b) = (2, -0.5)`
The point
`P(x,y)`
The directrix, c at
`y=-1.5`

The steps to find the equation of the parabola are as follows

Step 1
Find the distance between the focus and the point P using Pythagoras. We have two coordinates;
`(2, -0.5)` and
`(x,y)`.
We need the vertical and horizontal distances to find the hypotenuse (the diagram is shown in the second diagram).
The distance between the focus and point P is given by

`\sqrt{ (x-a)^(2)+ (y-b)^(2) }`

Step 2
Find the distance between the point P to the directrix
`c`. It is a vertical distance between y and c, expressed as
`y-c`

Step 3
The equation of parabola is then given as

`\sqrt{ (x-a)^(2)+ (y-b)^(2) }`=
`y-c`

`(x-a)^(2)+ (y-b)^(2)= (y-c)^(2)` ⇒ substituting a, b and c

`(x-2)^(2)+ (y--0.5)^(2) = (y--1.5)^(2)`

`(x-2)^(2)+ (y+0.5)^(2)= (y+1.5)^(2)`⇒Rearranging and making
`y` the subject gives


`y= ( x^(2) )/(2) -2x+1`