Question

derive the equation of a parabola with a focus at (10, -2) and a directrix at y = 6. enter your answers in the boxes. be sure to include the correct signs. y = □(x □)^2 □

Explanation:

Step1: Recall the definition of a parabola

The distance from a point $(x,y)$ on the parabola to the focus $(10, - 2)$ is equal to the distance from the point $(x,y)$ to the directrix $y = 6$.
The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, and the distance from the point $(x,y)$ to the line $y = k$ is $|y - k|$.
The distance from $(x,y)$ to $(10,-2)$ is $\sqrt{(x - 10)^2+(y + 2)^2}$, and the distance from $(x,y)$ to the line $y=6$ is $|y - 6|$. So, $\sqrt{(x - 10)^2+(y + 2)^2}=|y - 6|$.

Step2: Square both sides

Squaring both sides of the equation $\sqrt{(x - 10)^2+(y + 2)^2}=|y - 6|$, we get $(x - 10)^2+(y + 2)^2=(y - 6)^2$.
Expand the equations: $(x - 10)^2+y^{2}+4y + 4=y^{2}-12y + 36$.

Step3: Simplify the equation

Cancel out $y^{2}$ on both sides: $(x - 10)^2+4y + 4=-12y + 36$.
Move the terms with $y$ to one - side: $4y+12y=36 - 4-(x - 10)^2$.
$16y=32-(x - 10)^2$.

Step4: Solve for $y$

Divide both sides by 16: $y =-\frac{1}{16}(x - 10)^2+2$.

Answer:

$y=-\frac{1}{16}(x - 10)^2+2$