Question

(a) the equation of the circle in standard form is ( x^2 + y^2 = 49 ) (type an equation. simplify your answer.)
(b) the equation of the circle in standard form is (square). (type an equation. simplify your answer.)

Explanation:

Step1: Identify center and radius

The center of the circle is at \((5, 0)\) (from the diagram, the center is on the x - axis at \(x = 5\), \(y=0\)). A point on the circle is \((3, 0)\). The radius \(r\) is the distance between the center \((5,0)\) and the point \((3,0)\) on the circle. Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \((x_1,y_1)=(5,0)\) and \((x_2,y_2)=(3,0)\), we have \(r=\sqrt{(3 - 5)^2+(0 - 0)^2}=\sqrt{(- 2)^2}=2\).

Step2: Use standard circle equation

The standard form of the equation of a circle with center \((h,k)\) and radius \(r\) is \((x - h)^2+(y - k)^2=r^2\). Here, \(h = 5\), \(k = 0\) and \(r = 2\). Substituting these values into the formula, we get \((x - 5)^2+(y - 0)^2=2^2\), which simplifies to \((x - 5)^2+y^2 = 4\).

Answer:

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