(4pt) Petermine an equation for the parabola with focus $$ (-2,4) $$ and directrix $$ y=10 $$. The of the grid below is optional.)

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UpStudy AI · Accepted Answer

We start with the definition of a parabola: it is the set of all points $$ (x,y) $$ that are equidistant from the focus and the directrix. Given the focus $$ (-2,4) $$ and directrix $$ y=10 $$, a point $$ (x,y) $$ on the parabola satisfies

$$
\sqrt{(x+2)^2+(y-4)^2} = |y-10|.
$$

### Step 1. Square both sides

Squaring both sides of the equation to eliminate the square root gives

$$
(x+2)^2 + (y-4)^2 = (y-10)^2.
$$

### Step 2. Expand and Simplify

Expand the terms:

$$
\begin{aligned}
(x+2)^2 &= x^2 + 4x + 4, \
(y-4)^2 &= y^2 - 8y + 16, \
(y-10)^2 &= y^2 - 20y + 100.
\end{aligned}
$$

Substitute these into the equation:

$$
x^2 + 4x + 4 + y^2 - 8y + 16 = y^2 - 20y + 100.
$$

### Step 3. Cancel and Rearrange Terms

Cancel $$ y^2 $$ from both sides:

$$
x^2 + 4x - 8y + 20 = -20y + 100.
$$

Bring all terms to one side:

$$
x^2 + 4x - 8y + 20 + 20y - 100 = 0,
$$

which simplifies to

$$
x^2 + 4x + 12y - 80 = 0.
$$

### Step 4. Write in Standard Form

To express the parabola in standard form, complete the square for the $$ x $$-terms:

1. Group the $$ x $$-terms:

$$
   x^2 + 4x = (x^2 + 4x + 4) - 4 = (x+2)^2 - 4.
   $$

2. Substitute back into the equation:

$$
   (x+2)^2 - 4 + 12y - 80 = 0.
   $$

3. Combine the constant terms:

$$
   (x+2)^2 + 12y - 84 = 0.
   $$

4. Solve for $$ y $$:

$$
   12y = 84 - (x+2)^2,
   $$
   
   $$
   y = 7 - \frac{1}{12}(x+2)^2.
   $$

Alternatively, the standard (vertex) form of a vertical parabola is

$$
(x - h)^2 = 4p(y-k),
$$

where $$ (h,k) $$ is the vertex and $$ p $$ is the distance from the vertex to the focus. The vertex lies midway between the focus $$ (-2,4) $$ and the directrix $$ y=10 $$. The $$ y $$-coordinate of the vertex is

$$
k = \frac{4+10}{2} = 7,
$$

and its $$ x $$-coordinate is the same as that of the focus $$ h=-2 $$. The distance $$ p $$ is

$$
p = 4 - 7 = -3 \quad (\text{since the parabola opens downward}).
$$

Thus, the equation in vertex form is

$$
(x+2)^2 = -12(y-7).
$$

### Final Answer

$$
\boxed{(x+2)^2 = -12(y-7)}
$$
The equation of the parabola is $$ (x+2)^2 = -12(y-7) $$.

(4pt) Petermine an equation for the parabola with focus \( (-2,4) \) and directrix \( y=10 \). The of the grid below is optional.)

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