Question

A parabola can be drawn given a focus of (1,4) and a directrix of y=−2. Write the equation of the parabola in any form?

Answer 1

`y = 1/12x^2-1/6x+13/12`

Explanation:

The distance from the focus `(1,4)` to any point `(x,y)` on the parabola is:

`d = sqrt((x-1)^2+(y-4)^2)" [1]"`

The distance from the directrix `y = -2` to any point `(x,y)` on the parabola is:

`d = sqrt((y- -2)^2)" [2]"`

Because a parabola is defined as the locus of points equidistant from it focus and its directrix, we may set the right side of equation [1] equal to the right side of equation [2]:

`sqrt((y- -2)^2) = sqrt((x-1)^2+(y-4)^2)`

Square both sides:

`(y- -2)^2 = (x-1)^2+(y-4)^2`

Expand the squares:

`y^2+4y+4 = x^2-2x+1 + y^2 -8y+16`

Combine like terms:

`12y = x^2-2x + 13`

`y = 1/12x^2-1/6x+13/12`