Question

Math

Posted 6 months ago

```
Write the equation for a parabola with a focus at $(-4,3)$ and a directrix at $y=5$.
$y=$
$\square$
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

Identify the focus and directrix of the parabola. The focus is given as $(-4, 3)$ and the directrix is the line $y = 5$

step 2

Recall the definition of a parabola. A parabola is the set of all points that are equidistant from the focus and the directrix

step 3

Determine the vertex of the parabola. The vertex lies midway between the focus and directrix. Since the directrix is $y = 5$ and the focus has a $y$-coordinate of $3$, the vertex $y$-coordinate is $\frac{5+3}{2} = 4$. The $x$-coordinate of the vertex is the same as the focus, which is $-4$. Thus, the vertex is $(-4, 4)$

step 4

Write the standard form of the equation for a vertical parabola with vertex $(h, k)$: $(x-h)^2 = 4p(y-k)$, where $p$ is the distance from the vertex to the focus or directrix

step 5

Calculate the value of $p$. The distance from the vertex to the focus or directrix is $1$, since the vertex is at $y = 4$ and the directrix is at $y = 5$

step 6

Substitute the vertex coordinates and the value of $p$ into the standard form equation. We get $(x - (-4))^2 = 4(1)(y - 4)$

step 7

Simplify the equation to find the equation of the parabola. $(x + 4)^2 = 4(y - 4)$

Answer

The equation of the parabola is $(x + 4)^2 = 4(y - 4)$.

Key Concept

Standard form of a parabola's equation

Explanation

The standard form of a parabola's equation is derived from the definition of a parabola, using the distance from the vertex to the focus or directrix to determine the value of $p$ in the equation $(x-h)^2 = 4p(y-k)$.

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