Analytical Geometry Question Which Uses Distance Formula

Question

Show that the points P(–5, 0), Q(–2, 3), and R(6, –11) lie on a circle with its center at C(2, 4). Conclude by explaining why your findings show this.

Answer 1

(x-2)^2 + (y-4)^2 = r^2\ where r= radius. (2,4) is the circle's center

plug in the points

(6-2)^2 + (-11-4)^2= 16 + 225 = 241 = r^2

(-2-2)^2 + (3-4)^2 = 16 + 1 = 17

(-5-2)^2 + (0-4)^2 = 49+16= 65

they don't lie on a circle with center (2,4). the radius doesn't fit even 2 of the points, much less all 3 Just plot the points on a graph. They aren't part of the same circle

But, if you changed the circle center to (2,-4) those 3 points are on a circle

(6-2)^2 +(-11+4)^2 = 16+49= 65

(-2-2)^2 + (3+4)^2 = 16+49=65

they're all on a circle with radius = sqr65) = slightly over 8

Or more directly use the distance formula from each of the 3 points to the center point

d^2 =sum of squares of difference in y values & difference in corresponding x values

you get sqr65 for each which is the radius of the circle.