Question

Find an equation of the circle with center (-3, 8) and which passes through (7, -2)

Answer 1

we know the center is at (-3 , 8), and that it passes through (7 , -2), well so the distance from the center to (7 , -2) must be its radius.

`~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{-3}~,~\stackrel{y_1}{8})\qquad (\stackrel{x_2}{7}~,~\stackrel{y_2}{-2})\qquad \qquad d = √(( x_2- x_1)^2 + ( y_2- y_1)^2) \\\\\\ \stackrel{radius}{r}=√((~~7 - (-3)~~)^2 + (~~-2 - 8~~)^2)\implies r=√((7 +3)^2 + (-2 -8)^2) \\\\\\ r=√( (10)^2 + (-10)^2) \implies r=√( 100 + 100)\implies r=√( 200 ) \\\\[-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}{8})} \qquad \stackrel{radius}{\underset{r}{√(200)}} \\\\[-0.35em] ~\dotfill\\\\( ~~ x - (-3) ~~ )^2 ~~ + ~~ ( ~~ y-8 ~~ )^2~~ = ~~(√(200))^2\implies (x +3)^2 + (y -8)^2 = 200`