Question

What is the general form of the equation of a circle with its center at (-2, 1) and passing through (-4, 1)? A. X2 y2 − 4x 2y 1 = 0 B. X2 y2 4x − 2y 1 = 0 C. X2 y2 4x − 2y 9 = 0 D. X2 − y2 2x y 1 = 0.

Answer 1

well, we know it's centered at (-2 , 1) and it passes through (-4 , 1), if it passes through there, that means that the distance from the center to (-4 , 1) must be its radius.

$~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{-2}~,~\stackrel{y_1}{1})\qquad (\stackrel{x_2}{-4}~,~\stackrel{y_2}{1})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{[-4 - (-2)]^2 + [1 - 1]^2}\implies d=\sqrt{(-4+2)^2+0^2}\implies d=2$

$\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \qquad center~~(\stackrel{-2}{ h},\stackrel{1}{ k})\qquad \qquad radius=\stackrel{2}{ r} \\\\\\\ [x-(-2)]^2+[y-1]^2=2^2\implies (x+2)^2+(y-1)^2=4$