`(5/2,0)` Write the standard form of the equation of the parabola with the given focus and vertex at (0,0)
A parabola opens toward to the location of focus with respect to the vertex.
When the vertex and focus has same y-values, it implies that the parabola opens sideways (left or right).
When the vertex and focus has same x-values, it implies that the parabola may opens upward or downward.
The given focus of the parabola `(5/2,0)` is located at the right side of the vertex (0,0). Both points has the same value of `y=0` .
Thus, the parabola opens sideways towards to the right side of the vertex. In this case, we follow the standard formula: `(y-k)^2=4p(x-h).` We consider the following properties:
vertex as `(h,k) `
focus as `(h+p, k) `
directrix as `x=h-p`
Note: `p` is the distance of between focus and vertex or distance between directrix and vertex.
From the given vertex point `(0,0)` , we determine `h =0 ` and `k=0` .
From the given focus `(5/2,0)` , we determine `h+p =5/2 ` and `k=0` .
Applying `h=0` on `h+p=5/2` , we get:
Plug-in the values: `h=0` ,`k=0` , and `p=5/2` on the standard formula, we get:
`y^2=10x ` as the standard form of the equation of the parabola with vertex `(0,0)` and focus `(5/2,0)` .