To convert the given equation of the circle [tex]\( x^2 + 4x + y^2 - 10y + 13 = 0 \)[/tex] to its standard form, we need to complete the square for both the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms.
1. Completing the square for the [tex]\( x \)[/tex] terms:
The expression [tex]\( x^2 + 4x \)[/tex] can be transformed by completing the square.
- Take the coefficient of [tex]\( x \)[/tex], which is 4.
- Divide it by 2 to get 2, and then square it to get 4.
- So, [tex]\( x^2 + 4x \)[/tex] can be rewritten as [tex]\( (x + 2)^2 - 4 \)[/tex].
2. Completing the square for the [tex]\( y \)[/tex] terms:
The expression [tex]\( y^2 - 10y \)[/tex] can also be transformed by completing the square.
- Take the coefficient of [tex]\( y \)[/tex], which is -10.
- Divide it by 2 to get -5, and then square it to get 25.
- Thus, [tex]\( y^2 - 10y \)[/tex] can be rewritten as [tex]\( (y - 5)^2 - 25 \)[/tex].
3. Rewriting the original equation:
Substitute these completed squares back into the original equation:
[tex]\[
x^2 + 4x + y^2 - 10y + 13 = 0
\][/tex]
becomes:
[tex]\[
(x + 2)^2 - 4 + (y - 5)^2 - 25 + 13 = 0
\][/tex]
4. Combining constants:
Simplify the constants on the left side:
[tex]\[
(x + 2)^2 + (y - 5)^2 - 16 = 0
\][/tex]
Add 16 to both sides to isolate the squared terms:
[tex]\[
(x + 2)^2 + (y - 5)^2 = 16
\][/tex]
Thus, the standard form of the equation of the circle is:
[tex]\[
(x + 2)^2 + (y - 5)^2 = 16
\][/tex]
Therefore, the correct option is:
[tex]\((x+2)^2+(y-5)^2=16\)[/tex]
The correct answer is:
[tex]\[
\boxed{(x+2)^2+(y-5)^2=16}
\][/tex]
