Question

Which equation represents a parabola with a focus of (-2,5) and a directrix of y=9 regent?

Answer 1

When the distance from the vertex to the focus is "p" and the vertex is (h, k), the equation of the parabola can be written as
  y = (1/(4p))(x -h)² +k

Of course, p is half the distance between the focus and the directrix, so is
  (1/2)(5 - 9) = -4/2 = -2
The vertex is (-2, 7), so the equation of the parabola can be written as
  y = (-1/8)(x +2)² +7

Answer 2

The equation of the parabola with a focus at $\left( { - 2,5} \right)$ is $\boxed{y = - \frac{1}{8}{{\left( {x + 2} \right)}^2} + 7}$.

Further Explanation:

The standard form of the parabola is shown below.

$\boxed{y = \frac{1}{{4a}}{{\left( {x - h} \right)}^2} + k}$

Here, the parabola has vertex at $\left( {h,k} \right)$ and has the symmetry parallel to x-axis and it opens left.

Given:

The focus of the parabola is $\left( { - 2,5} \right).$

The directrix of the parabola is $y = 9.$

Calculation:

Vertex lies in the middle of focus and directrix.

The value of a can be obtained as follows,

$\begin{aligned}a &= \frac{1}{2}\left( {5 - 9} \right)\\&= \frac{{ - 4}}{2}\\&= - 2\\\end{aligned}$

The vertex of the parabola is $\left( { - 2,7} \right).$

The equation of the parabola can be obtained as follows,

$y = \dfrac{{ - 1}}{8}{\left( {x + 2} \right)^2} + 7$

The equation of the parabola with a focus at $\left( { - 2,5} \right)$ is $\boxed{y = - \frac{1}{8}{{\left( {x + 2} \right)}^2} + 7}.$

Learn more:

1. Learn more about unit conversion https://brainly.com/question/4837736
2. Learn more about non-collinear https://brainly.com/question/4165000
3. Learn more about binomial and trinomial https://brainly.com/question/1394854

Answer details:

Grade: High School

Subject: Mathematics

Chapter: Conic sections

Keywords: vertex, symmetry, symmetric, axis, y-axis, x-axis, function, graph, parabola, focus, vertical parabola, upward parabola, downward parabola.