A circle is drawn on a coordinate plane with the center point at (−8,−2) and a radius of 2. Derive the equation of the circle in standard form, using the given information and the Pythagorean Theorem.(1 point)

Question

Responses

(x+2)2+(y+8)2=2
left parenthesis x plus 2 right parenthesis squared plus left parenthesis y plus 8 right parenthesis squared equals 2

(x−8)2+(y−2)2=4
left parenthesis x minus 8 right parenthesis squared plus left parenthesis y minus 2 right parenthesis squared equals 4

(x+8)2+(y+2)2=4
left parenthesis x plus 8 right parenthesis squared plus left parenthesis y plus 2 right parenthesis squared equals 4

(x−2)2+(y−8)2=2
left parenthesis x minus 2 right parenthesis squared plus left parenthesis y minus 8 right parenthesis squared equals 2

Answer 1

To derive the equation of a circle in standard form, we use the formula:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

In this case, the center is given as \((-8, -2)\) and the radius is \(2\).

1. Identify the center coordinates:

   • \(h = -8\)
   • \(k = -2\)

2. Since the radius \(r = 2\), we calculate \(r^2\):

   • \(r^2 = 2^2 = 4\)

3. Substitute \(h\), \(k\), and \(r^2\) into the standard form equation: \[ (x - (-8))^2 + (y - (-2))^2 = 4 \] This simplifies to: \[ (x + 8)^2 + (y + 2)^2 = 4 \]

So the equation of the circle in standard form is: \[ (x + 8)^2 + (y + 2)^2 = 4 \]

The correct response from the given options is: \[ (x + 8)^2 + (y + 2)^2 = 4 \]