`(5/2,0)` Write the standard form of the equation of the parabola with the given focus and vertex at (0,0)

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marizi eNotes educator| Certified Educator

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:

`0+p=5/2`

`p=5/2` 

Plug-in the values: `h=0` ,`k=0` , and `p=5/2` on the standard formula, we get: 

`(y-0)^2=4*5/2(x-0)` 

`y^2=10x `  as the standard form of the equation of the parabola with vertex `(0,0)` and focus `(5/2,0)` .