Question

Use the given information to write an equation for each parabola.

d
The parabola has focus (2, –2) and directrix x=12.

Answer 1

`(y+2)^2=-20(x-7)`

Explanation:

The directrix of a parabola is a fixed line outside of the parabola that is perpendicular to the axis of symmetry. Therefore, as the directrix is vertical (x = 12), the parabola has a horizontal axis of symmetry. This means that the parabola is horizontal (opens left or right).

The focus of a parabola is a fixed point located inside the parabola. Therefore, as the focus (2, -2) is to the left of the directrix, the parabola opens to the left. The axis of symmetry of the given parabola is the y-value of the focus, so the axis of symmetry is y = -2.

The vertex is located on the axis of symmetry, therefore its y-value is y = -2. Since all points on the parabola are equidistant to both the focus and the directrix, the x-value of the vertex is the midpoint of the x-values of the focus (x = 2) and the directrix (x = 12). Therefore, the x-value of the vertex is:

`x=(12+2)/(2)=7`

So, the vertex (h, k) is (7, -2).

The standard form of a parabola with a horizontal axis of symmetry is:

`\boxed{(y-k)^2=4p(x-h)}`

where:

• p ≠ 0
• Vertex = (h, k)
• Focus = (h+p, k)
• Directrix: x = (h - p)
• Axis of symmetry: y = k

The value of p is the distance between the focus and the directrix. As the parabola opens to the left, the value of p will be negative. Therefore:

`-|p|=-\left|(2-12)/(2)\right|=-|5|=-5`

Substitute the values of h, k and p into the standard formula:

`(y-(-2))^2=4(-5)(x-7)`

Simplify:

`(y+2)^2=-20(x-7)`

So, the equation for the parabola with focus (2, -2) and directrix x = 12 is:

`\large\boxed{\boxed{(y+2)^2=-20(x-7)}}`

`\hrulefill`

Additional notes

To write the equation in the standard form of x = ay² + by + c, rearrange the equation to isolate x:

`(y+2)^2=-20(x-7)`

`-(1)/(20)(y+2)^2=x-7`

`x=-(1)/(20)(y+2)^2+7`

`x=-(1)/(20)(y^2+4y+4)+7`

`x=-(1)/(20)y^2-(1)/(5)y+(34)/(5)`

Answer 2

`\sf y^2 + 20x + 4y - 136 =0`

Explanation:

The equation of a parabola with focus (h, k) and directrix x = p is:

`\sf (x - h)^2 + (y - k)^2 = (x - p)^2`

where

• (h, k) is the vertex of the parabola
• p is the focal length

In this case, we have:

• h = 2
• k = -2
• p = 12

Substitute these values into the formula, we get the following equation for the parabola:

`\sf (x - 2)^2 + (y - (-2))^2 = (x - 12)^2`

Expanding the squares, we get:

`\sf x^2 - 4x + 4 + y^2 + 4y + 4 = x^2 - 24x + 144`

Combining like terms, we get

`\sf x^2 - 4x + 4 + y^2 + 4y + 4 -x^2 + 24x - 144=0`

`\sf y^2 + 20x + 4y - 136 = 0`

Therefore, the equation of the parabola is:

`\sf y^2 + 20x + 4y - 136 = 0`