Question

1. $(x - 3)^2+(y + 1)^2=25$ part i: what kind of conic section is this? (2 points) part ii: what are the coordinates of its center? (2 points) part iii: identify one other critical characteristic of this graph and its value. (2 points)

Explanation:

Step1: Recall conic - section equations

The general equation of a circle is \((x - a)^2+(y - b)^2=r^2\), where \((a,b)\) is the center and \(r\) is the radius. The given equation \((x - 3)^2+(y+1)^2 = 25\) is in this form.

Step2: Find the center

Comparing \((x - 3)^2+(y+1)^2 = 25\) with \((x - a)^2+(y - b)^2=r^2\), we have \(a = 3\) and \(b=-1\). So the center is \((3,-1)\).

Step3: Find another characteristic

The radius \(r\) satisfies \(r^2 = 25\). Taking the square - root of both sides (and considering the non - negative value for radius), we get \(r = 5\).

Answer:

Part I: Circle
Part II: \((3,-1)\)
Part III: Radius, \(r = 5\)