Question

what is the equation of the circle in the diagram?
options:

• $(x - 4)^2 + (y - 3)^2 = 2$
• $(x - 3)^2 + (y - 4)^2 = 2$
• $(x - 3)^2 + (y - 4)^2 = 4$
• $(x - 4)^2 + (y - 3)^2 = 4$

Explanation:

Step1: Find the center of the circle

From the diagram, the center of the circle is at \((3, 4)\) (since it's 3 units along the x - axis and 4 units along the y - axis). The standard form of the equation of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center and \(r\) is the radius. So \(h = 3\) and \(k = 4\), which means the equation should start with \((x - 3)^{2}+(y - 4)^{2}\).

Step2: Find the radius of the circle

Looking at the circle, the distance from the center \((3,4)\) to the top (or bottom) of the circle. The center is at \(y = 4\), and the bottom of the circle is at \(y = 2\), so the radius \(r=4 - 2=2\). Then \(r^{2}=2^{2} = 4\).

Step3: Write the equation of the circle

Substituting \(h = 3\), \(k = 4\) and \(r^{2}=4\) into the standard form, we get \((x - 3)^{2}+(y - 4)^{2}=4\).

Answer:

\(\boldsymbol{(x - 3)^{2}+(y - 4)^{2}=4}\) (corresponding to the option \((x - 3)^{2}+(y - 4)^{2}=4\))