Question

derive the equation of a parabola with a focus at (-1, -4) and a directrix at y = -6. move one answer to each box. find the distance from a point (x, y) on the parabola to the focus. find the distance from (x, y) to the directrix. set up an equation between the distances, then square both sides

Explanation:

Step1: Use distance formula for point - focus distance

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, the focus is $(-1,-4)$ and the point on the parabola is $(x,y)$. So the distance $d_1=\sqrt{(x + 1)^2+(y + 4)^2}$.

Step2: Use distance formula for point - directrix distance

The directrix is $y=-6$. The distance from the point $(x,y)$ to the horizontal line $y = - 6$ is $d_2=\vert y+6\vert$.

Step3: Set up and simplify the equation

Since a parabola is defined as the set of points equidistant from the focus and the directrix, we set $d_1 = d_2$. So $\sqrt{(x + 1)^2+(y + 4)^2}=\vert y + 6\vert$. Squaring both sides gives $(x + 1)^2+(y + 4)^2=(y + 6)^2$. Expand the right - hand side: $(x + 1)^2+y^{2}+8y + 16=y^{2}+12y+36$. Simplify the equation: $(x + 1)^2=4y + 20$, or $(x + 1)^2=4(y + 5)$.

Answer:

Distance from point $(x,y)$ to focus: $\sqrt{(x + 1)^2+(y + 4)^2}$
Distance from point $(x,y)$ to directrix: $\vert y + 6\vert$
Equation after squaring: $(x + 1)^2+(y + 4)^2=(y + 6)^2$ (or simplified to $(x + 1)^2=4(y + 5)$)