Question

question
find the endpoints of the latus rectum of the parabola below.
$(x + 2)=\frac{1}{20}(y - 1)^2$
enter the endpoints as ordered pairs $(x,y)$.
provide your answer below:
the endpoints of the latus rectum are and .

Explanation:

Step1: Identify the standard - form and parameters

The given equation \((x + 2)=\frac{1}{20}(y - 1)^2\) is of the form \((x - h)=\frac{1}{4p}(y - k)^2\), where \((h,k)\) is the vertex of the parabola. Comparing, we have \(h=-2\), \(k = 1\) and \(\frac{1}{4p}=\frac{1}{20}\), so \(4p = 20\) and \(p=5\).

Step2: Find the endpoints of the latus - rectum

For a parabola of the form \((x - h)=\frac{1}{4p}(y - k)^2\), the length of the latus - rectum is \(|4p|\) and the endpoints of the latus - rectum have \(y\) - coordinates \(y=k\pm2p\) and \(x\) - coordinate \(x=h + p\).
Since \(h=-2\), \(k = 1\) and \(p = 5\), the \(x\) - coordinate of the endpoints of the latus - rectum is \(x=-2 + 5=3\).
The \(y\) - coordinates are \(y=1+2\times5=11\) and \(y=1-2\times5=-9\).

Answer:

\((3,11)\) and \((3, - 9)\)