Question: Complete the equation of the circle $x^2 + y^2 + 2x + 10y + 10 = 0$ in standard form, identify the center and radius, then select the correct graph of the circle.
$(x ? ____)^2 + (y ? ____)^2 = ____$
The center is C(____, ____) and the radius is r = ____.
A
B
C
D
position_hint: center
1. **State the problem:** We need to rewrite the circle equation $$x^2 + y^2 + 2x + 10y + 10 = 0$$ in standard form, find its center and radius.
2. **Recall the standard form of a circle:**
$$ (x - h)^2 + (y - k)^2 = r^2 $$
where $(h,k)$ is the center and $r$ is the radius.
3. **Group $x$ and $y$ terms:**
$$ x^2 + 2x + y^2 + 10y = -10 $$
4. **Complete the square for $x$ terms:**
Take half of the coefficient of $x$, which is $2$, half is $1$, square it: $1^2 = 1$.
Add and subtract $1$:
$$ x^2 + 2x + 1 - 1 $$
5. **Complete the square for $y$ terms:**
Take half of $10$, which is $5$, square it: $5^2 = 25$.
Add and subtract $25$:
$$ y^2 + 10y + 25 - 25 $$
6. **Rewrite the equation including these:**
$$ (x^2 + 2x + 1) + (y^2 + 10y + 25) = -10 + 1 + 25 $$
7. **Simplify:**
$$ (x + 1)^2 + (y + 5)^2 = 16 $$
8. **Identify center and radius:**
Center: $C(-1, -5)$
Radius: $r = \sqrt{16} = 4$
**Final answers:**
$$(x + 1)^2 + (y + 5)^2 = 16$$
Center: $(-1, -5)$
Radius: $4$