Question

find the focus of teh parabola that has vertex (-3,2), opens horizontally, and passes through the point (-10,1).

Answer 1

`(-(85)/(28),2)`

Explanation:

We are asked to find the focus of the parabola that has a vertex
`(-3,2)` opens horizontally and passes through the point
`(-10,1)`.

We know that equation of a horizontal parabola is
`(y-k)^2=4p(x-h)`, where p is not equal to zero.

`(h,k)` = Vertex of parabola.

Focus:
`(h+p,k)`

Upon substituting the coordinates of vertex and given point in equation of parabola, we will get:

`(1-2)^2=4p(-10-(-3))`

`(-1)^2=4p(-10+3)`

`1=4p(-7)`

`1=-28p`

`(1)/(-28)=(-28p)/(-28)`

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

Focus:
`(h+p,k)`

`(-3+(-(1)/(28)),2)`

`(-3-(1)/(28),2)`

`((-3*28)/(28)-(1)/(28),2)`

`((-84)/(28)-(1)/(28),2)`

`((-84-1)/(28),2)`

`((-85)/(28),2)`

`(-(85)/(28),2)`

Therefore, the focus of the parabola is at point
`(-(85)/(28),2)`.