Question

derive the equation of the parabola with a focus at (2, 4) and a directrix of y = 8. (2 points)
f(x)=-\frac{1}{8}(x - 2)^2+6
f(x)=\frac{1}{8}(x - 2)^2+6
f(x)=-\frac{1}{8}(x + 2)^2+8
f(x)=\frac{1}{8}(x + 2)^2+8

Explanation:

Step1: Recall the definition of a parabola

The distance from any point $(x,y)$ on the parabola to the focus $(2,4)$ is equal to the distance from the point $(x,y)$ to the directrix $y = 8$. 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 $|y - k|$. So, $\sqrt{(x - 2)^2+(y - 4)^2}=|y - 8|$.

Step2: Square both sides

Squaring both sides of the equation $\sqrt{(x - 2)^2+(y - 4)^2}=|y - 8|$, we get $(x - 2)^2+(y - 4)^2=(y - 8)^2$.

Step3: Expand the equations

Expand the right - hand side and left - hand side: $(x - 2)^2+y^{2}-8y + 16=y^{2}-16y + 64$.

Step4: Simplify the equation

Cancel out $y^{2}$ on both sides: $(x - 2)^2-8y + 16=-16y+64$.
Move the terms involving $y$ to one side: $-8y + 16y=64-(x - 2)^2 - 16$.
$8y=48-(x - 2)^2$.
$y=-\frac{1}{8}(x - 2)^2+6$. So, $f(x)=-\frac{1}{8}(x - 2)^2+6$.

Answer:

$f(x)=-\frac{1}{8}(x - 2)^2+6$