To find the equation of a circle given its center and radius, we start with the standard form of a circle's equation, which is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where \((h, k)\) is the center of the circle and \(r\) is the radius.
Given:
- The center of the circle \((h, k) = (3, 8)\)
- The radius \(r = 5\)
We substitute these values into the standard form:
[tex]\[
(x - 3)^2 + (y - 8)^2 = 5^2
\][/tex]
Simplify the radius squared:
[tex]\[
(x - 3)^2 + (y - 8)^2 = 25
\][/tex]
Next, we expand the squared binomials:
[tex]\[
(x - 3)^2 = x^2 - 6x + 9
\][/tex]
[tex]\[
(y - 8)^2 = y^2 - 16y + 64
\][/tex]
Putting it all together:
[tex]\[
x^2 - 6x + 9 + y^2 - 16y + 64 = 25
\][/tex]
Combine all the terms on the left side to form a single equation, moving the constant from the right side to the left:
[tex]\[
x^2 + y^2 - 6x - 16y + 9 + 64 - 25 = 0
\][/tex]
Simplify the constants:
[tex]\[
x^2 + y^2 - 6x - 16y + 48 = 0
\][/tex]
Thus, the general form of the equation for this circle is:
[tex]\[
x^2 + y^2 - 6x - 16y + 48 = 0
\][/tex]
Therefore, the correct answer is:
A. [tex]\(x^2 + y^2 - 6 x - 16 y + 48 = 0\)[/tex]
