Question

determine the following for the parabola given by the equation
y ^ { 2 } + 2 y - 20 x + 41 = 0
find the focus.
( x , y )
find an equation for the directrix.
directrix
find the endpoints of the latus rectum. enter your answer as ordered pairs separated by a
comma.
endpoints
graph the parabola using the focus and the directrix. to move the focus and directrix, drag
the black points.

Explanation:

Step1: Rewrite the equation in standard form

Complete the square for the \(y\) - terms.

$$
LATEXBLOCK0
$$

The standard form of a parabola of the form \((y - k)^{2}=4p(x - h)\)

Step2: Identify \(h,k,p\)

Comparing \((y + 1)^{2}=20(x - 2)\) with \((y - k)^{2}=4p(x - h)\)
We have \(h = 2,k=-1\), and \(4p=20\Rightarrow p = 5\)

Step3: Find the focus

The focus of the parabola \((y - k)^{2}=4p(x - h)\) is \((h + p,k)\)
Substitute \(h = 2,k=-1,p = 5\)
The focus is \((2 + 5,-1)=(7,-1)\)

Step4: Find the directrix

The directrix of the parabola \((y - k)^{2}=4p(x - h)\) is \(x=h - p\)
Substitute \(h = 2,p = 5\)
\(x=2-5=-3\)

Step5: Find the endpoints of the latus - rectum

The endpoints of the latus - rectum of the parabola \((y - k)^{2}=4p(x - h)\) are \((h + p,k\pm2p)\)
Substitute \(h = 2,k=-1,p = 5\)
\(x=h + p=7\), \(y=-1\pm10\)
The endpoints are \((7,9)\) and \((7,-11)\)

Answer:

• Focus: \((7,-1)\)
• Directrix: \(x=-3\)
• Endpoints of the latus rectum: \((7,9),(7,-11)\)