Question

the equation of a circle is given below. identify the center and the radius. then graph the circle.
$x^{2}+y^{2}+2x + 4y=11$
center: (-1, -2)
radius: 4

Explanation:

Step1: Complete the square for x - terms

\[x^{2}+2x=(x + 1)^{2}-1\]

Step2: Complete the square for y - terms

\[y^{2}+4y=(y + 2)^{2}-4\]

Step3: Rewrite the circle equation

The original equation \(x^{2}+y^{2}+2x + 4y=11\) becomes \((x + 1)^{2}-1+(y + 2)^{2}-4=11\), which simplifies to \((x + 1)^{2}+(y + 2)^{2}=11 + 1+4=16\)

Step4: Identify the center and radius

The standard form of a circle equation is \((x - a)^{2}+(y - b)^{2}=r^{2}\), where \((a,b)\) is the center and \(r\) is the radius. For \((x + 1)^{2}+(y + 2)^{2}=16\), the center is \((-1,-2)\) and \(r = 4\)

Answer:

Center: \((-1,-2)\)
Radius: \(4\)