Question

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

Answer 1

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

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

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

Given:

• Center: \((h, k) = (-8, -2)\)
• Radius: \(r = 2\)

Now we can plug these values into the formula:

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

2. Substitute the radius \(r = 2\) in the equation: \[ (x + 8)^2 + (y + 2)^2 = 2^2 \] This gives: \[ (x + 8)^2 + (y + 2)^2 = 4 \]

Thus, the equation of the circle in standard form is:

\[ \boxed{(x + 8)^2 + (y + 2)^2 = 4} \]