Answer :
The equation of a circle in standard form is given by:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
Given:
- The center of the circle is [tex]\((-7, 9)\)[/tex]. Therefore, [tex]\(h\)[/tex] is [tex]\(-7\)[/tex] and [tex]\(k\)[/tex] is [tex]\(9\)[/tex].
- The radius of the circle is [tex]\(5\)[/tex].
To complete the equation of the circle, we can substitute the given values into the standard form equation.
1. Substitute the center coordinates [tex]\((h, k) = (-7, 9)\)[/tex] into the equation:
[tex]\[ (x - (-7))^2 + (y - 9)^2 = r^2 \][/tex]
Simplifying [tex]\((x - (-7))\)[/tex] gives us [tex]\((x + 7)\)[/tex]:
[tex]\[ (x + 7)^2 + (y - 9)^2 = r^2 \][/tex]
2. Substitute the radius [tex]\(r = 5\)[/tex] into the equation:
[tex]\[ (x + 7)^2 + (y - 9)^2 = 5^2 \][/tex]
3. Compute the square of the radius:
[tex]\[ 5^2 = 25 \][/tex]
Therefore, the completed equation of the circle is:
[tex]\[ (x + 7)^2 + (y - 9)^2 = 25 \][/tex]
So, the boxed parts in your question can be filled in as follows:
\[
(x - [-7])^2 + (y - [9])^2 = [25]
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
Given:
- The center of the circle is [tex]\((-7, 9)\)[/tex]. Therefore, [tex]\(h\)[/tex] is [tex]\(-7\)[/tex] and [tex]\(k\)[/tex] is [tex]\(9\)[/tex].
- The radius of the circle is [tex]\(5\)[/tex].
To complete the equation of the circle, we can substitute the given values into the standard form equation.
1. Substitute the center coordinates [tex]\((h, k) = (-7, 9)\)[/tex] into the equation:
[tex]\[ (x - (-7))^2 + (y - 9)^2 = r^2 \][/tex]
Simplifying [tex]\((x - (-7))\)[/tex] gives us [tex]\((x + 7)\)[/tex]:
[tex]\[ (x + 7)^2 + (y - 9)^2 = r^2 \][/tex]
2. Substitute the radius [tex]\(r = 5\)[/tex] into the equation:
[tex]\[ (x + 7)^2 + (y - 9)^2 = 5^2 \][/tex]
3. Compute the square of the radius:
[tex]\[ 5^2 = 25 \][/tex]
Therefore, the completed equation of the circle is:
[tex]\[ (x + 7)^2 + (y - 9)^2 = 25 \][/tex]
So, the boxed parts in your question can be filled in as follows:
\[
(x - [-7])^2 + (y - [9])^2 = [25]
