Question

3 the equation ( x^2 + y^2 - 4x + 2y = b ) describes a circle. part a determine the ( y )-coordinate of the center of the circle. enter your answer in the box. part b the radius of the circle is 7 units. what is the value of ( b ) in the equation? enter your answer in the box.

Explanation:

Part A

Step1: Complete the square for \(x\) and \(y\) terms

The general equation of a circle is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius.
Given \(x^{2}+y^{2}-4x + 2y=b\).
Group \(x\) - terms and \(y\) - terms: \((x^{2}-4x)+(y^{2}+2y)=b\).
For the \(x\) - terms: \(x^{2}-4x=(x - 2)^{2}-4\) (using \((a - b)^2=a^{2}-2ab + b^{2}\), here \(a = x\), \(b = 2\)).
For the \(y\) - terms: \(y^{2}+2y=(y + 1)^{2}-1\) (using \((a + b)^2=a^{2}+2ab + b^{2}\), here \(a = y\), \(b = 1\)).
So the equation becomes \((x - 2)^{2}-4+(y + 1)^{2}-1=b\), or \((x - 2)^{2}+(y + 1)^{2}=b + 5\).

Step2: Identify the \(y\) - coordinate of the center

Comparing with \((x - h)^2+(y - k)^2=r^2\), the center is \((h,k)=(2,-1)\). The \(y\) - coordinate of the center \(k=-1\).

Part B

Step1: Use the radius formula

Since the radius \(r = 7\) and from \((x - 2)^{2}+(y + 1)^{2}=b + 5\), we know that \(r^{2}=b + 5\).
Substitute \(r = 7\) into \(r^{2}=b + 5\). Then \(7^{2}=b+5\).

Step2: Solve for \(b\)

\(49=b + 5\).
Subtract 5 from both sides: \(b=49 - 5\).

Answer:

• Part A: \(-1\)
• Part B: \(44\)