Answer :
Step-by-step explanation:
To derive the equation of a parabola given the focus \((-4, -3)\) and the directrix \(y = 1\), we use the definition of a parabola: it is the set of all points \((x, y)\) such that the distance to the focus is equal to the distance to the directrix.
1. **Distance to the Focus**: The distance between any point \((x, y)\) on the parabola and the focus \((-4, -3)\) is given by the distance formula:
\[
\sqrt{(x + 4)^2 + (y + 3)^2}
\]
2. **Distance to the Directrix**: The distance between any point \((x, y)\) on the parabola and the directrix \(y = 1\) is:
\[
|y - 1|
\]
According to the definition of the parabola:
\[
\sqrt{(x + 4)^2 + (y + 3)^2} = |y - 1|
\]
3. **Square Both Sides**: To eliminate the square root, we square both sides of the equation:
\[
(x + 4)^2 + (y + 3)^2 = (y - 1)^2
\]
4. **Expand Both Sides**:
\[
(x + 4)^2 + (y + 3)^2 = y^2 - 2y + 1
\]
\[
x^2 + 8x + 16 + y^2 + 6y + 9 = y^2 - 2y + 1
\]
5. **Simplify**: Combine like terms and simplify:
\[
x^2 + 8x + 16 + y^2 + 6y + 9 = y^2 - 2y + 1
\]
\[
x^2 + 8x + 16 + 6y + 9 = -2y + 1
\]
\[
x^2 + 8x + 25 + 6y = -2y + 1
\]
6. **Combine \(y\) Terms**:
\[
x^2 + 8x + 25 + 6y + 2y = 1
\]
\[
x^2 + 8x + 8y + 25 = 1
\]
7. **Isolate the \(y\) Terms**:
\[
x^2 + 8x + 8y + 25 - 25 = 1 - 25
\]
\[
x^2 + 8x + 8y = -24
\]
\[
8y = -x^2 - 8x - 24
\]
\[
y = -\frac{1}{8}x^2 - x - 3
\]
So, the equation of the parabola in standard form is:
\[
y = -\frac{1}{8}x^2 - x - 3
\]
