Math / Geometry

QuestionQuestion 4 of 10 This quiz: 10 poir This question: 1
Find the equation of the parabola with focus $(-3,6)$ and directrix $y=4$ .
Choose the correct equation below. $(x-3)^{2}=-4(y-5)$ $(x-3)^{2}=4(y-5)$ $(x+3)^{2}=4(y-5)$ $(x+3)^{2}=-4(y-5)$

Studdy Solution

STEP 1

1. The focus of the parabola is at the point $(-3, 6)$ .
2. The directrix of the parabola is the line $y = 4$ .
3. The parabola is vertical since the directrix is horizontal.

STEP 2

1. Determine the vertex of the parabola.
2. Calculate the distance $p$ from the vertex to the focus (or directrix).
3. Write the equation of the parabola using the vertex form for a vertical parabola.

STEP 3

Determine the vertex of the parabola:
The vertex is midway between the focus and the directrix. The y-coordinate of the vertex is the average of the y-coordinate of the focus and the y-coordinate of the directrix.
$y_{\text{vertex}} = \frac{6 + 4}{2} = 5$
The x-coordinate of the vertex is the same as the x-coordinate of the focus since the parabola is vertical.
$x_{\text{vertex}} = -3$
Thus, the vertex is at $(-3, 5)$ .

STEP 4

Calculate the distance $p$ from the vertex to the focus (or directrix):
$p = 6 - 5 = 1$
Since the focus is above the directrix, the parabola opens upwards, and $p$ is positive.

STEP 5

Write the equation of the parabola using the vertex form for a vertical parabola:
The standard form of a parabola with vertex $(h, k)$ and distance $p$ is:
$(x - h)^2 = 4p(y - k)$
Substitute $h = -3$ , $k = 5$ , and $p = 1$ :
$(x + 3)^2 = 4(1)(y - 5)$ $(x + 3)^2 = 4(y - 5)$
The correct equation of the parabola is:
$\boxed{(x + 3)^2 = 4(y - 5)}$

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