Question

derive the equation of a parabola with a focus at (-1, -7) and a directrix at y = -1. enter your answers in the boxes. be sure to include the correct signs. y = □(x □)^2□

Explanation:

Step1: Recall the distance formula

The distance between a point $(x,y)$ on the parabola and the focus $(-1,-7)$ is $\sqrt{(x + 1)^2+(y + 7)^2}$. The distance between the point $(x,y)$ and the directrix $y=-1$ is $|y + 1|$. Since the distance from any point on the parabola to the focus is equal to the distance from that point to the directrix, we have $\sqrt{(x + 1)^2+(y + 7)^2}=|y + 1|$.

Step2: Square both sides

Squaring both sides of the equation $\sqrt{(x + 1)^2+(y + 7)^2}=|y + 1|$ gives us $(x + 1)^2+(y + 7)^2=(y + 1)^2$.

Step3: Expand the equation

Expand the squares: $(x + 1)^2+y^{2}+14y + 49=y^{2}+2y+1$.

Step4: Simplify the equation

Cancel out $y^{2}$ on both sides: $(x + 1)^2+14y + 49=2y+1$. Then, move the terms involving $y$ to one - side: $14y-2y=1 - 49-(x + 1)^2$.

Step5: Solve for $y$

$12y=-48-(x + 1)^2$. So, $y=-\frac{1}{12}(x + 1)^2-4$.

Answer:

$y=-\frac{1}{12}(x + 1)^2-4$