Find an equation of the parabola described and state the two points that define the latus rectum.
-Focus at (0, 4); directrix the line y = -4
A) $y ^ { 2 } = 4 x$; latus rectum: $( 9,2 )$ and $( - 9,2 )$
B) $x ^ { 2 } = 4 y$; latus rectum: $( 2,4 )$ and $( - 2,4 )$
C) $x ^ { 2 } = 16 y$; latus rectum: $( 8,4 )$ and $( - 8,4 )$
D) $x ^ { 2 } = 16 y$; latus rectum: $( 4,8 )$ and $( - 4,8 )$

Question

Find an equation of the parabola described and state the two points that define the latus rectum.
-Focus at (0, 4); directrix the line y = -4 
A) $y ^ { 2 } = 4 x$; latus rectum: $( 9,2 )$ and $( - 9,2 )$
B) $x ^ { 2 } = 4 y$; latus rectum: $( 2,4 )$ and $( - 2,4 )$
C) $x ^ { 2 } = 16 y$; latus rectum: $( 8,4 )$ and $( - 8,4 )$
D) $x ^ { 2 } = 16 y$; latus rectum: $( 4,8 )$ and $( - 4,8 )$

Quizplus · Accepted Answer

The focus is at (0, 4) and the directrix is y = -4, which means the vertex is halfway between the focus and directrix, at (0, 0). The parabola opens upwards because the focus is above the directrix. The distance between the focus and directrix is 8, so the equation is $x^2 = 4py$, where $p = 8/2 = 4$, giving $x^2 = 16y$. The length of the latus rectum is $4p = 16$, and since the vertex is at the origin and the parabola opens upwards, the latus rectum is horizontal, passing through the focus. Thus, its endpoints are $(8, 4)$ and $(-8, 4)$.

Find an Equation of the Parabola Described and State the Two