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: Recall the definition of a parabola

A point $(x,y)$ lies on a parabola with focus $(x_0,y_0)$ and directrix $y = k$ if the distance from the point $(x,y)$ to the focus is equal to the distance from the point $(x,y)$ to the directrix. 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 a point $(x,y)$ to the horizontal line $y = k$ is $|y - k|$. Here, $(x_0,y_0)=(0,2)$ and $k=-2$. So, $\sqrt{(x - 0)^2+(y - 2)^2}=|y+2|$.

Step2: Square both sides of the equation

Squaring both sides of $\sqrt{x^{2}+(y - 2)^2}=|y + 2|$ gives $x^{2}+(y - 2)^2=(y + 2)^2$. Expand the squares: $x^{2}+y^{2}-4y + 4=y^{2}+4y+4$. Simplify the equation: $x^{2}=8y$.

Step3: Check each point

For point A $(-4,2)$

Substitute $x=-4$ and $y = 2$ into $x^{2}=8y$. We get $(-4)^{2}=16$ and $8\times2 = 16$. So, $(-4,2)$ lies on the parabola.

For point B $(0,0)$

Substitute $x = 0$ and $y=0$ into $x^{2}=8y$. We get $0^{2}=0$ and $8\times0=0$. So, $(0,0)$ lies on the parabola.

For point C $(0,2)$

Substitute $x = 0$ and $y = 2$ into $x^{2}=8y$. We get $0^{2}=0$ and $8\times2=16$. So, $(0,2)$ does not lie on the parabola.

For point D $(4,3)$

Substitute $x = 4$ and $y = 3$ into $x^{2}=8y$. We get $4^{2}=16$ and $8\times3 = 24$. So, $(4,3)$ does not lie on the parabola.

For point E $(8,8)$

Substitute $x = 8$ and $y = 8$ into $x^{2}=8y$. We get $8^{2}=64$ and $8\times8=64$. So, $(8,8)$ lies on the parabola.

Answer:

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