Question

Any point on the parabola can be labeled (x,y), as shown. What are the distances from the point (x,y) to the focus of the parabola and the directrix? Select two answers.distance to the focus: (squareroot over this whole problem)* (x+3)^2+(y-3)^2distance to the directix: |y-4|distance to the directix: |y+4|distance to the focus: *squareroot over again* (x+3)^2+(y-2)^2distance to the directix: |x-4|distance to the focus: *square root again* (x-2)^2+(y+3)^2

Answer 1

Given:

Vertex: (-3, 3)

Focus: (-3, 2)

Let's find the distance from the point (x, y) to the focus of the parabola and the directrix.

To find the distance, apply the distance formula:

`d=√((x_2-x_1)^2+(y_2-y_1)^2)`

Thus, we have:

Distance from (x, y) to the focus:

Where:

(x1, y1) ==> (x, y)

(x2, y2) ==> (-3, 2)

Thus, we have:

`\begin{gathered} d=√((x-(-3))^2+(y-2)^2) \\ \\ d=√((x+3)^2+(y-2)^2) \end{gathered}`

Therefore, the distance from the point (x, y) to the focus is:

`√((x+3)^2+(y-2)^2)`

• The distance from the point to the directrix.

From the graph, the directrix is:

y = 4

Now, to find the distance, subtract the y-coordinate of the point from y = 4.

The distance is the absolute value of the result.

Thus, we have:

`|y-4|`

ANSWER:

Distance from the point to the focus:

`√((x+3)^2+(y-2)^2)`

Distance from the point to directrix:

`|y-4|`