Question

PLSSSSS HELP!!! WHY IS NOBODY ANSWERING!! PARABOLAS

Answer 1

1.
`x=0`

2.
`(x+5)^2=16(y-2)`

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

4. See explanation

5.
`(y-6)^2=-4(x-3)`

Explanation:

1. The equation of given parabola is

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

The vertex of this parabola is at point (1,-3), this parabola goes in positive x-direction and has the parameter
`p,` such that

`2p=4\\ \\p=2`

The directrix is the vertical line. The equation of the directrix is

`x-x_0=-(p)/(2)\ [x_0\text{ is the x-coordinate of the vertex}]\\ \\x-1=-(2)/(2)\\ \\x-1=-1\\ \\x=0`

2. The vertex of the parabola is at point (-5,2), the focus is at point (-5,6), this means that the vertex and the focus both lie on the vertical line x = -5. This line is the line of symmetry of the parabola. Thus, parabola goes in positive y-direction and has the equation

`(x-(-5))^2=2p(y-2)\\ \\(x+5)^2=2p(y-2)`

To find the parameter
`p,` note that the distance between vertex and focus is equal to
`(p)/(2),` hence,

`(p)/(2)=|6-2|\\ \\(p)/(2)=4\\ \\p=8`

and the equation of the parabola is

`(x+5)^2=2\cdot 8\cdot (y-2)\\ \\(x+5)^2=16(y-2)`

3. Given the equation

`3x^2+24x-2y+52=0`

Rewrite this equation as follows

`3(x^2+8x)-2y+52=0\\ \\3(x^2+2\cdot 4x+4^2-4^2)=2y-52\\ \\3((x+4)^2-16)=2y-52\\ \\3(x+4)^2-48=2y-52\\ \\3(x+4)^2=2y-52+48\\ \\3(x+4)^2=2y-4\\ \\3(x+4)^2=2(y-2)\\ \\(x+4)^2=(2)/(3)(y-2)`

4. Given the equation of the parabola
`(y+1)^2=12(x-3)`

The vertex of the parabola is at point (3,-1).

The parabola opens to the right.

Parabola's parameter is
`p=(12)/(2)=6`, so the focus is
`(p)/(2)=(6)/(2)=3`units away from the vertex.

The directrix is
`p=6` units away from the focus.

The focus is at the point
`(3+3,-1)=(6,-1)` and the directrix has the equation

`x-3=-3\\ \\x=0`

5. The distance between the directrix and the focus is

`p=|4-2|=2` units.

The directrix is to the right from the focus, so parabola opens to the left and has the equation

`(y-y_0)^2=-2p(x-x_0),`

where
`(x_0,y_0)` is the vertex.

The vertex is the point that is half way from the focus to the directrix, hence

`x_0=(4+2)/(2)=3\\ \\y_0=6`

So, the equation is

`(y-6)^2=-4(x-3)`