Question

write the standard equation for each of the circles in parts (a) through (c). the coordinates of the center and the radius for each circle are integers. (a) the equation of the circle in standard form is \\(\square\\). (type an equation. simplify your answer.)

Explanation:

Step1: Find the center of the circle

From the diagram, the center of the circle is at the origin \((0,0)\) (since the center is the intersection of the horizontal and vertical lines passing through \((-7,0)\) and \((0, -7)\) etc., and it's the midpoint, which is \((0,0)\)).

Step2: Find the radius of the circle

The distance from the center \((0,0)\) to a point on the circle, say \((-7,0)\), is the radius. Using the distance formula \(r=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), with \((x_1,y_1)=(0,0)\) and \((x_2,y_2)=(-7,0)\), we get \(r = \sqrt{(-7 - 0)^2+(0 - 0)^2}=\sqrt{49}=7\).

Step3: Write the standard equation of the circle

The standard form of the equation of a circle is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius. Substituting \(h = 0\), \(k = 0\), and \(r = 7\), we get \((x - 0)^2+(y - 0)^2=7^2\), which simplifies to \(x^2+y^2 = 49\).

Answer:

\(x^2 + y^2 = 49\)