Question

1. select the equation for a graph that is the set of all points in the plane that are equidistant from the point f(0, 5) and line y = -5

options:
$y = \frac{1}{8}x^2$
$y = \frac{1}{20}x^2$
$y = -\frac{1}{20}x^2$
$y = -\frac{1}{8}x^2$

Explanation:

Step1: Define general point $(x,y)$

Let $(x,y)$ be any point on the graph.

Step2: Distance to focus $F(0,5)$

Use distance formula:
$\sqrt{(x-0)^2 + (y-5)^2}$

Step3: Distance to line $y=-5$

Vertical distance:
$|y - (-5)| = |y+5|$

Step4: Set distances equal

Equate the two distances:
$\sqrt{x^2 + (y-5)^2} = |y+5|$

Step5: Square both sides

Eliminate square root and absolute value:
$x^2 + (y-5)^2 = (y+5)^2$

Step6: Expand both sides

Expand the squared terms:
$x^2 + y^2 -10y +25 = y^2 +10y +25$

Step7: Simplify the equation

Cancel like terms and solve for $y$:
$x^2 -10y = 10y$
$x^2 = 20y$
$y = \frac{1}{20}x^2$

Answer:

$\boldsymbol{y = \frac{1}{20}x^2}$ (corresponding to the third option)