Question

select all the points which lie on the parabola with focus at (0, 2) and directrix at y = -2. a. (-4, 2) b. (0, 0) c. (0, 2) d. (4, 3) e. (8, 8)

Explanation:

Step1: Derive parabola equation

A parabola is the set of points where the distance to the focus equals the distance to the directrix. For a point $(x,y)$, distance to focus $(0,2)$ is $\sqrt{(x-0)^2+(y-2)^2}$, distance to directrix $y=-2$ is $|y - (-2)| = |y+2|$. Set equal and square both sides:
$$\sqrt{x^2+(y-2)^2} = |y+2|$$
$$x^2+(y-2)^2=(y+2)^2$$
Expand both sides:
$$x^2 + y^2 -4y +4 = y^2 +4y +4$$
Simplify to get:
$$x^2 = 8y \quad \text{or} \quad y=\frac{x^2}{8}$$

Step2: Test point A (-4,2)

Substitute $x=-4$ into $y=\frac{x^2}{8}$:
$$y=\frac{(-4)^2}{8}=\frac{16}{8}=2$$
This matches the y-coordinate of the point.

Step3: Test point B (0,0)

Substitute $x=0$ into $y=\frac{x^2}{8}$:
$$y=\frac{0^2}{8}=0$$
This matches the y-coordinate of the point.

Step4: Test point C (0,2)

Substitute $x=0$ into $y=\frac{x^2}{8}$:

$$y=\frac{0^2}{8}=0 eq 2$$

This does not match.

Step5: Test point D (4,3)

Substitute $x=4$ into $y=\frac{x^2}{8}$:

$$y=\frac{4^2}{8}=\frac{16}{8}=2 eq 3$$

This does not match.

Step6: Test point E (8,8)

Substitute $x=8$ into $y=\frac{x^2}{8}$:
$$y=\frac{8^2}{8}=\frac{64}{8}=8$$
This matches the y-coordinate of the point.

Answer:

A. $(-4, 2)$, B. $(0, 0)$, E. $(8, 8)$