Find the standard form of the equation of the parabola with the given characteristics. $\text { Vertex: } ( 1,3 ) \text {; directrix: } y = - 2$
A) $( x - 1 ) ^ { 2 } = 3 ( y - 3 )$
B) $( x - 1 ) ^ { 2 } = 20 ( y - 3 )$
C) $( y - 1 ) ^ { 2 } = 3 ( x - 3 )$
D) $( x - 1 ) ^ { 2 } = - 20 ( y - 3 )$
E) $( y - 1 ) ^ { 2 } = - 20 ( x - 3 )$

Question

Find the standard form of the equation of the parabola with the given characteristics. $\text { Vertex: } ( 1,3 ) \text {; directrix: } y = - 2$ 
A) $( x - 1 ) ^ { 2 } = 3 ( y - 3 )$
B) $( x - 1 ) ^ { 2 } = 20 ( y - 3 )$
C) $( y - 1 ) ^ { 2 } = 3 ( x - 3 )$
D) $( x - 1 ) ^ { 2 } = - 20 ( y - 3 )$
E) $( y - 1 ) ^ { 2 } = - 20 ( x - 3 )$

Quizplus · Accepted Answer

Since the vertex is given as $(1,3)$ and the directrix is given as $y=-2$, the parabola opens upward. 
The focus is located at $(1,3+p)$ where $p$ is the distance between the vertex and the directrix. In this case, $p=5$. So the focus is at $(1,8)$. 
Using the distance formula, we can set up an equation to find the points on the parabola:

$\sqrt{(x-1)^2+(y-8)^2}=|y+2|$

Squaring both sides and simplifying, we get:

$(x-1)^2=20(y-3)$

Therefore, the standard form of the equation of the parabola is $(x-1)^2=20(y-3)$.

Find the Standard Form of the Equation of the Parabola Vertex: (1,3); directrix: y=−2\text { Vertex: } ( 1,3 ) \text {; directrix: } y = - 2 Vertex: (1,3); directrix: y=−2

Find the Standard Form of the Equation of the Parabola $\text { Vertex: } ( 1,3 ) \text {; directrix: } y = - 2$