Question

Question #52596

Answer 1

`y = 1/16x^2-1/4x+5/4`

Explanation:

For the second parabola we translate the focus 1 unit right and 2 units up:

`(1+1, 3+2) = (2,5)`

The directrix cannot be translated horizontally but it can be translated up:

`y = -5+2`

`y = -3`

The distance from the focus `(2,5)` to any point `(x,y)` on the second parabola is:

`d = sqrt((x-2)^2+(y-5)^2)" [1]"`

The distance from the directrix, `y = -3` to any point `(x,y)` on the second parabola is:

`d = sqrt((y - (-3))^2)`

Change the `--` to `+`:

`d = sqrt((y +3)^2)" [2]"`

A parabola is the locus of points equidistant from its focus and its directrix, therefore, we may set the right side of equation [1] equal to the right side of equation [2]:

`sqrt((y +3)^2) = sqrt((x-2)^2+(y-5)^2)`

Square both sides:

`(y +3)^2 = (x-2)^2+(y-5)^2`

Expand the squares:

`y^2 + 6y + 9 = x^2-4x+4+y^2-10y+25`

Combine like terms:

`16y = x^2-4x + 20`

`y = 1/16x^2-1/4x+5/4`