Question

Using the given information, give the vertex form equation of each parabola.

Vertex: (4,-3) Focus:(47/12,-3)

Answer 1

The equation of parabola is given by :
`(x-4) = (-1)/(3)(y+3)^(2)`

Explanation:

Given that vertex and focus of parabola are

Vertex: (4,-3)

Focus:(
`(47)/(12)`,-3)

The general equation of parabola is given by.

`(x-h)^(2) = 4p(y-k)`, When x-componet of focus and Vertex is same

`(x-h) = 4p(y-k)^(2)`, When y-componet of focus and Vertex is same

where Vertex: (h,k)

and p is distance between vertex and focus

The distance between two points is given by :

L=
`\sqrt{(X2-X1)^(2)+(Y2-Y1)^(2)}`

For value of p:

p=
`\sqrt{(X2-X1)^(2)+(Y2-Y1)^(2)}`

p=
`\sqrt{(4-(47)/(12))^(2)+((-3)-(-3))^(2)}`

p=
`\sqrt{((1)/(12))^(2)}`

p=
`(1)/(12)` and p=
`(-1)/(12)`

Since, Focus is left side of the vertex,

p=
`(-1)/(12)` is required value

Replacing value in general equation of parabola,

Vertex: (h,k)=(4,-3)

p=
`(-1)/(12)`

`(x-h) = 4p(y-k)^(2)`

`(x-4) = 4((-1)/(12))(y+3)^(2)`

`(x-4) = (-1)/(3)(y+3)^(2)`