Question

write the equation of the circle centered at $(-9,-7)$ that passes through $(19,3)$.

Explanation:

Step1: Find the radius

The radius \(r\) of the circle is the distance between the center \((-9,-7)\) and the point \((19,3)\) on the circle. Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Here \(x_1=-9,y_1 = - 7,x_2=19,y_2 = 3\).
\(r=\sqrt{(19-(-9))^2+(3-(-7))^2}=\sqrt{(19 + 9)^2+(3 + 7)^2}=\sqrt{(28)^2+(10)^2}=\sqrt{784 + 100}=\sqrt{884}\)

Step2: Write the 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 of the circle.
Given \(h=-9,k=-7\) and \(r^2 = 884\)
The equation is \((x+9)^2+(y + 7)^2=884\)

Answer:

\((x + 9)^2+(y + 7)^2=884\)