Question

A circle has the center of (1,-5) and a radius of 5 determine the location of the point (4,-1)

Answer 1

"determine the location" or namely, is it inside the circle, outside the circle, or right ON the circle?

well, we know the center is at (1,-5) and it has a radius of 5, so the distance from the center to any point on the circle will just be 5, now if (4,-1) is less than that away, is inside, if more than that is outiside and if it's exactly 5 is right ON the circle.

well, we can check by simply getting the distance from the center to the point (4,-1).

`\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ \stackrel{center}{(\stackrel{x_1}{1}~,~\stackrel{y_1}{-5})}\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{-1})\qquad \qquad d = √(( x_2- x_1)^2 + ( y_2- y_1)^2) \\\\\\ d = √([4-1]^2+[-1-(-5)]^2)\implies d=√((4-1)^2+(-1+5)^2) \\\\\\ d = √(3^2+4^2)\implies d =√(9+16)\implies d=√(25)\implies \stackrel{\textit{right on the circle}}{d = 5}`