Question

the equation of a circle is given below. identify the radius and the center. then graph the circle.\\(4x^{2}+4y^{2}+16x - 8y - 101 = 0\\)

Explanation:

Step1: Divide by 4 to simplify

Divide the entire equation \(4x^{2}+4y^{2}+16x - 8y-101 = 0\) by 4:
\(x^{2}+y^{2}+4x - 2y-\frac{101}{4}=0\)

Step2: Group x and y terms

Group the \(x\)-terms and \(y\)-terms:
\((x^{2}+4x)+(y^{2}-2y)=\frac{101}{4}\)

Step3: Complete the square for x

For \(x^{2}+4x\), add \((\frac{4}{2})^{2}=4\) to both sides:
\((x^{2}+4x + 4)+(y^{2}-2y)=\frac{101}{4}+4\)

Step4: Complete the square for y

For \(y^{2}-2y\), add \((\frac{-2}{2})^{2}=1\) to both sides:
\((x^{2}+4x + 4)+(y^{2}-2y + 1)=\frac{101}{4}+4 + 1\)

Step5: Simplify both sides

Left side becomes \((x + 2)^{2}+(y - 1)^{2}\).
Right side: \(\frac{101}{4}+5=\frac{101 + 20}{4}=\frac{121}{4}\).

The standard circle equation is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center and \(r\) is the radius.
Here, \(h=-2\), \(k = 1\), and \(r^{2}=\frac{121}{4}\), so \(r=\frac{11}{2}\).

Answer:

Radius: \(\frac{11}{2}\)
Center: \((-2, 1)\)