Question

What is the equation of the circle with Center (-6, 7) that passes through the point (4, -2) ​

Answer 1

we know the center of the circle, and we also know a point on the circle, well, the distance from the center to a point is just the radius.

`\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{-6}~,~\stackrel{y_1}{7})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{-2})\qquad \qquad d = √(( x_2- x_1)^2 + ( y_2- y_1)^2) \\\\\\ \stackrel{radius}{r}=√([4-(-6)]^2+[-2-7]^2)\implies r=√((4+6)^2+(-2-7)^2) \\\\\\ r=√(10^2+(-9)^2)\implies r=√(100+81)\implies r=√(181) \\\\[-0.35em] ~\dotfill`

`\bf \textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \qquad center~~(\stackrel{-6}{ h},\stackrel{7}{ k})\qquad \qquad radius=\stackrel{√(181)}{ r}\\[2em] [x-(-6)]^2+[y-7]^2=(√(181))^2\implies (x+6)^2+(y-7)^2=181`