Answer :
To find the equation of a circle given its center and radius, we can use the standard form of the equation of a circle:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where \((h, k)\) represents the coordinates of the center of the circle, and \(r\) is the radius.
In this problem, we are given:
- The center of the circle \((h, k)\) is \((3, -2)\)
- The radius \(r\) is \(5\)
Substitute the given values into the standard equation:
1. Substitute \(h = 3\) and \(k = -2\):
[tex]\[ (x - 3)^2 + (y - (-2))^2 = r^2 \][/tex]
2. Simplify the equation, noting that \(y - (-2)\) is the same as \(y + 2\):
[tex]\[ (x - 3)^2 + (y + 2)^2 = r^2 \][/tex]
3. Substitute \(r = 5\):
[tex]\[ (x - 3)^2 + (y + 2)^2 = 5^2 \][/tex]
4. Calculate \(5^2\):
[tex]\[ (x - 3)^2 + (y + 2)^2 = 25 \][/tex]
Thus, the equation of the circle with center \((3, -2)\) and radius \(5\) is:
[tex]\[ (x - 3)^2 + (y + 2)^2 = 25 \][/tex]
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where \((h, k)\) represents the coordinates of the center of the circle, and \(r\) is the radius.
In this problem, we are given:
- The center of the circle \((h, k)\) is \((3, -2)\)
- The radius \(r\) is \(5\)
Substitute the given values into the standard equation:
1. Substitute \(h = 3\) and \(k = -2\):
[tex]\[ (x - 3)^2 + (y - (-2))^2 = r^2 \][/tex]
2. Simplify the equation, noting that \(y - (-2)\) is the same as \(y + 2\):
[tex]\[ (x - 3)^2 + (y + 2)^2 = r^2 \][/tex]
3. Substitute \(r = 5\):
[tex]\[ (x - 3)^2 + (y + 2)^2 = 5^2 \][/tex]
4. Calculate \(5^2\):
[tex]\[ (x - 3)^2 + (y + 2)^2 = 25 \][/tex]
Thus, the equation of the circle with center \((3, -2)\) and radius \(5\) is:
[tex]\[ (x - 3)^2 + (y + 2)^2 = 25 \][/tex]
