To write the equation of a circle in standard form, we follow these steps:
1. Identify the center (h, k) of the circle:
The center is given as [tex]\((5, -7)\)[/tex].
2. Identify a point on the circle:
A point on the circle is given as [tex]\((-3, -1)\)[/tex].
3. Calculate the radius [tex]\( r \)[/tex]:
The radius is the distance from the center to the given point on the circle. The distance [tex]\( r \)[/tex] can be determined using the distance formula:
[tex]\[
r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}
\][/tex]
Substituting the given values:
[tex]\[
r = \sqrt{(-3 - 5)^2 + (-1 + 7)^2}
\][/tex]
[tex]\[
r = \sqrt{(-8)^2 + 6^2}
\][/tex]
[tex]\[
r = \sqrt{64 + 36}
\][/tex]
[tex]\[
r = \sqrt{100}
\][/tex]
[tex]\[
r = 10
\][/tex]
4. Write the equation of the circle in standard form:
The standard form of a circle's equation with center [tex]\((h, k)\)[/tex] and radius [tex]\( r \)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Substituting the center [tex]\((5, -7)\)[/tex] and radius [tex]\( 10 \)[/tex]:
[tex]\[
(x - 5)^2 + (y + 7)^2 = 10^2
\][/tex]
[tex]\[
(x - 5)^2 + (y + 7)^2 = 100
\][/tex]
Therefore, the equation of the circle in standard form is:
[tex]\[
(x - 5)^2 + (y + 7)^2 = 100
\][/tex]
