Question

Find the focus and directrix of the parabola `y=2a(x-a)^2`?

Answer 1

The focus is: `(a, 1/(8a))`

The directrix is `y = -1/(8a)`

Explanation:

The given equation `y=2a(x-a)^2` is in the vertex form:

`y= 1/(4f)(x-h)^2+k`

where, the vertex is `(h,k)`, the focus is `(h,k+f)`, and the equation of the directrix is `y=k-f`

By observation, we can write the following equations:

`2a = 1/(4f)" [1]"`

`h = a" [2]"`

`k = 0" [3]"`

Write equation [1] as `f` in terms of `a`:

`f = 1/(8a)" [1.1]"`

Use equations, [1.1], [2], and [3] to write the focus:

`(h,k+f) = (a, 0+1/(8a))`

Simplify:

`(h,k+f) = (a, 1/(8a))`

Use equations [3] and [1.1] to write the equation of the directrix:

`y = 0- 1/(8a)`

Simplify:

`y = -1/(8a)`