The line segment that has endpoints on a parabola, passes through the focus of the parabola, and is perpendicular to the axis of symmetry is called the latus rectum of the parabola. Find the length of the latus rectum for the parabola .
A)152
B)38
C)76
D)57
E)19

Question

The line segment that has endpoints on a parabola, passes through the focus of the parabola, and is perpendicular to the axis of symmetry is called the latus rectum of the parabola.  Find the length of the latus rectum for the parabola  .
A)152 
B)38 
C)76 
D)57 
E)19

Quizplus · Accepted Answer

Given that the parabola equation is $x^{2}=16y$. The focus of the parabola is at $(0,4)$, which is $\frac{1}{4}$ units above the vertex.
As the line segment is perpendicular to the axis of symmetry, it must be parallel to the y-axis.
Let the equation of the line segment be $x=k$, where k is a constant. As $x=k$ passes through the focus $(0,4)$, we get $k=0$.
Thus, the line segment passes through $(0,4)$ and is parallel to the y-axis. To find the length of the latus rectum, we must find the distance from the focus to the directrix, which is $4$ units.
Therefore, the length of the latus rectum is $2	imes4=\boxed{	extbf{(C)}\ 76}$.

The Line Segment That Has Endpoints on a Parabola, Passes