Question

given a circle with the equation $x^{2}+y^{2}+2x + 14y+30 = 0$, what is the coordinate of its center and the length of its radius? enter your answers in the boxes. the coordinate of the circles center is (□,□). the circles radius is □ units.

Explanation:

Step1: Rewrite the circle equation in standard form

The general equation of a circle is \(x^{2}+y^{2}+2gx + 2fy+c = 0\), and its standard - form is \((x + g)^{2}+(y + f)^{2}=r^{2}\), where the center is \((-g,-f)\) and the radius \(r=\sqrt{g^{2}+f^{2}-c}\). Given the equation \(x^{2}+y^{2}+2x + 14y+30 = 0\), we have \(2g = 2\), so \(g = 1\); \(2f=14\), so \(f = 7\); and \(c = 30\).

Step2: Find the center of the circle

The center of the circle \((x + g)^{2}+(y + f)^{2}=r^{2}\) is \((-g,-f)\). Substituting \(g = 1\) and \(f = 7\), the center is \((-1,-7)\).

Step3: Calculate the radius of the circle

The radius \(r=\sqrt{g^{2}+f^{2}-c}\). Substitute \(g = 1\), \(f = 7\), and \(c = 30\) into the formula: \(r=\sqrt{1^{2}+7^{2}-30}=\sqrt{1 + 49-30}=\sqrt{20}=2\sqrt{5}\).

Answer:

The coordinate of the circle's center is \((-1,-7)\). The circle's radius is \(2\sqrt{5}\) units.