Question

Determine the equation of the circle with the center

Answer 1

since the distance from the center of a circle to a point on the circle is by definition its radius, so the distance from those two points above will be the radius, let's get that.

`~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{3}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{-3}~,~\stackrel{y_2}{-√(13)})\qquad \qquad d = √(( x_2- x_1)^2 + ( y_2- y_1)^2) \\\\\\ \stackrel{ radius }{r}=\sqrt{(~~-3 - 3~~)^2 + (~~-√(13) - 0~~)^2}\implies r=\sqrt{ (-6)^2 + (-√(13))^2} \\\\\\ r=√( 36 + 13)\implies r=√( 49 )\implies r=7 \\\\[-0.35em] ~\dotfill`

`\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{h}{3}~~,~~\underset{k}{0})} \qquad \stackrel{radius}{\underset{r}{7}} \\\\[-0.35em] ~\dotfill\\\\( ~~ x - 3 ~~ )^2 ~~ + ~~ ( ~~ y-0 ~~ )^2~~ = ~~7^2\implies (x -3)^2 + y^2 = 49`