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−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

(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

Answer 1

To derive the equation of the circle in standard form, we start with the general formula for the equation of a circle, which is given by:

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

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

Given:

• The center of the circle is \((-8, -2)\), so \(h = -8\) and \(k = -2\).
• The radius is \(2\), so \(r = 2\).

Now, we plug the values into the equation:

1. Substitute \(h\) and \(k\): \[ (x - (-8))^2 + (y - (-2))^2 = r^2 \]

   Which simplifies to: \[ (x + 8)^2 + (y + 2)^2 = 2^2 \]

2. Calculate \(r^2\): \[ 2^2 = 4 \]

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

Therefore, the correct representation from the given options is:

\((x + 8)^2 + (y + 2)^2 = 4\) or in another format: (x + 8)^2 + (y + 2)^2 = 4.