Question

What is an equation of a parabola with the given vertex and focus vertex (0 0) focus (2.5, 0)?

Answer 1

We most often deal with vertical parabolas (open up or down) and horizontal parabolas (open right or left). For this type of parabolas vertex and focus lie on same vertical or horizontal line. Parabola opens up to the side that contains focus. If focus is left of vertex parabola opens to left.

If we have horizontal parabola formula is:
$(y-k)^{2} =4a(x-h)$
If we have vertical parabola formula is:
$(x-h)^{2} =4a(y-k)$

In this problem we have:
vertex (0,0)
focus (2.5,0)

We can see that focus is right of vertex so parabola opens to right. It is horizontal parabola. We will use formula:
$(y-k)^{2} =4a(x-h)$
Where:
vertex(h,k) = (0,0)
a = distance between vertex and focus:
$a= \sqrt{ ( x_{2}- x_{1} )^{2}+( y_{2}- y_{1} )^{2} } = \sqrt{( 0- 2.5 )^{2}+( 0-0)^{2}} =2.5$

Now we can insert these numbers:
$(y-0)^{2} =4*2.5(x-0)$
$y^{2} =10x$