Question

write an equation of a circle with diameter ab.
a(2,2), b(14,14)
choose the correct answer below.
oa. (x - 8)^2 + (y - 8)^2 = 72
ob. (x + 8)^2 + (y - 8)^2 = 72
oc. (x - 2)^2 - (y - 2)^2 = 14
od. (x + 8)^2 - (y + 8)^2 = 72
oe. (x + 14)^2 - (y + 14)^2 = 2
of. (x - 8)^2 + (y - 8)^2 = 14
og. (x - 8)^2 + (y - 8)^2 = 2
oh. (x - 8)^2 - (y + 8)^2 = 72

Explanation:

Step1: Find the center of the circle

The center of the circle is the mid - point of the diameter.
The mid - point formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
For \(A(2,2)\) and \(B(14,14)\), the center \((h,k)=(\frac{2 + 14}{2},\frac{2+14}{2})=(8,8)\).

Step2: Find the radius of the circle

The radius \(r\) is half of the length of the diameter.
The distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(A(2,2)\) and \(B(14,14)\), \(d=\sqrt{(14 - 2)^2+(14 - 2)^2}=\sqrt{12^2+12^2}=\sqrt{144 + 144}=\sqrt{288}=12\sqrt{2}\).
So \(r = \frac{d}{2}=6\sqrt{2}\), and \(r^{2}=(6\sqrt{2})^{2}=72\).

Step3: Write the equation of the circle

The standard form of the equation of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\).
Substituting \(h = 8,k = 8,r^{2}=72\), we get \((x - 8)^{2}+(y - 8)^{2}=72\).

Answer:

A. \((x - 8)^{2}+(y - 8)^{2}=72\)