Question

find the endpoints of the latus rectum of the parabola below. (x - 2)^2=-2(y + 2) select the correct answer below: the endpoints of the latus rectum are (6, -5/2) and (-2, -5/2). the endpoints of the latus rectum are (-5/2, 6) and (-5/2, -2). the endpoints of the latus rectum are (-5/2, 3) and (-5/2, 1). the endpoints of the latus rectum are (3, -5/2) and (1, -5/2). the endpoints of the latus rectum are (8, -5/2) and (-4, -5/2). the endpoints of the latus rectum are (-5/2, 8) and (-5/2, -4).

Explanation:

Step1: Identify the form of the parabola

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

Step2: Find the focus of the parabola

The focus of the parabola \((x - h)^2 = 4p(y - k)\) is \((h,k + p)\). Substituting \(h = 2\), \(k=-2\) and \(p =-\frac{1}{2}\), we get the focus \((2,-2-\frac{1}{2})=(2,-\frac{5}{2})\).

Step3: Find the endpoints of the latus - rectum

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

Answer:

The endpoints of the latus rectum are \((3,-\frac{5}{2})\) and \((1,-\frac{5}{2})\).