Question

write an equation of a circle with diameter ab.
a(1,1), b(7,7)
choose the correct answer below.
oa. (x + 4)^2 - (y + 4)^2 = 18
oc. (x - 4)^2 - (y + 4)^2 = 18
oe. (x - 4)^2 + (y - 4)^2 = 1
og. (x - 4)^2 + (y - 4)^2 = 7
ob. (x + 7)^2 - (y + 7)^2 = 1
od. (x - 1)^2 - (y - 1)^2 = 7
of. (x + 4)^2 + (y - 4)^2 = 18
oh. (x - 4)^2 + (y - 4)^2 = 18

Explanation:

Step1: Find the center of the circle

The center of the circle is the mid - point of the diameter \(AB\).
The mid - point formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
For \(A(1,1)\) and \(B(7,7)\), the center \((h,k)=(\frac{1 + 7}{2},\frac{1+7}{2})=(4,4)\).

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 \(A(1,1)\) and \(B(7,7)\), \(d=\sqrt{(7 - 1)^2+(7 - 1)^2}=\sqrt{36 + 36}=\sqrt{72}=6\sqrt{2}\).
So \(r = \frac{d}{2}=\frac{6\sqrt{2}}{2}=3\sqrt{2}\), and \(r^{2}=(3\sqrt{2})^{2}=18\).

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 \(h = 4,k = 4,r^{2}=18\), we get \((x - 4)^2+(y - 4)^2=18\).

Answer:

H. \((x - 4)^2+(y - 4)^2=18\)