Question

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

Answer 1

Focus is `((3a)/2,0)` and directrix is `2x=a`

Explanation:

The equation of parabola is `y^2=2a(x-a)`

or `2ax=y^2+2a^2`

or `2a(x-a)=y^2`

In a horizontal parabola of the form `4p(x-h)=(y-k)^2`, the vertex is `(h,k)`, focus is `(h+p,k)` and directrix is `x=h-p`

Here `p=(2a)/4=a/2`, `h=a` and `k=0`

Therefore its vertex is `(a,0)`, focus is `(a+a/2,0)` or `((3a)/2,0)`

and directrix is `x=a/2` or `2x=a`

If `a=4`, the parabola, focus and directrix appear as

graph{(y^2-8x+32)((x-6)^2+y^2-0.01)(x-2)=0 [-5, 15, -5, 5]}