Question

use the graph below to determine the equation of the circle in (a) center - radius form and (b) general form. a. type the equation in center - radius form (simplify your answer.) b. type the equation in general form (simplify your answer.)

Explanation:

Step1: Find the center of the circle

The center of the circle is the mid - point between any two points on the circle that are diametrically opposite. The points \((0,4)\) and \((10,4)\) are diametrically opposite. The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). For \((x_1 = 0,y_1=4)\) and \((x_2 = 10,y_2 = 4)\), the center \((h,k)=(\frac{0 + 10}{2},\frac{4+4}{2})=(5,4)\).

Step2: Calculate the radius

The radius \(r\) is the distance between the center \((5,4)\) and a point on the circle, say \((5,9)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), with \((x_1 = 5,y_1 = 4)\) and \((x_2=5,y_2 = 9)\), we have \(r=\sqrt{(5 - 5)^2+(9 - 4)^2}=\sqrt{0 + 25}=5\).

Step3: Write the center - radius form

The center - radius form of a circle is \((x - h)^2+(y - k)^2=r^2\). Substituting \(h = 5,k = 4,r = 5\) gives \((x - 5)^2+(y - 4)^2=25\).

Step4: Expand to general form

Expand \((x - 5)^2+(y - 4)^2=25\). \((x - 5)^2=x^{2}-10x + 25\) and \((y - 4)^2=y^{2}-8y+16\). So \(x^{2}-10x + 25+y^{2}-8y + 16=25\), which simplifies to \(x^{2}+y^{2}-10x-8y + 16=0\).

Answer:

a. \((x - 5)^2+(y - 4)^2=25\)
b. \(x^{2}+y^{2}-10x-8y + 16=0\)