Question

write the standard form of the equation of the circle. 17)

Explanation:

Step1: Find the center of the circle

The center of the circle \((h,k)\) is the mid - point of the diameter. The mid - point formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Given the endpoints of the diameter \((x_1,y_1)=(3,6)\) and \((x_2,y_2)=(7,6)\).

$$h=\frac{3 + 7}{2}=\frac{10}{2}=5,y=\frac{6+6}{2}=6$$

So, the center of the circle is \((h,k)=(5,6)\)

Step2: Find the radius of the circle

The radius \(r\) is the distance from the center \((h,k)\) to one of the endpoints of the diameter. Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Let \((x_1,y_1)=(5,6)\) and \((x_2,y_2)=(7,6)\)

$$r=\sqrt{(7 - 5)^2+(6 - 6)^2}=\sqrt{2^2+0^2}=\sqrt{4}=2$$

Step3: Write the standard form of the equation of the circle

The standard form of the equation of a circle is \((x - h)^2+(y - k)^2=r^2\). Substituting \(h = 5,k = 6,r = 2\)
\((x - 5)^2+(y - 6)^2=4\)

Answer:

\((x - 5)^2+(y - 6)^2=4\)