Find the equation of a circle containing the points (7,5), (1,-7), and (9,-1).

Question

PLEASE NEED HELP RIGHT NOW! THIS TOPIC IS ABOUT CONICS

Answer 1

There are a few ways to do this. Here is one way (probably the easiest).

Let (x,y) be the coordinates of the center. Let R = the radius of the circle.

There are then 3 equations and 3 unknowns.

(x-7)²+(y-5)²=R²

(x-1)²+(y+7)²=R²

(x-9)²+(y+1)²=R²

Since all left sides equal R², the left sides must be equal.

Set two pairs of left sides equal to each other and come up with two equations and two unknowns. Thankfully the x² and y² terms disappear. Solve for x and y giving the coordinates of the center. Plug those values into one of the original equations to find R.

Answer 2

(x-h)^2 + (y-k)^2 = r^2 where (h,k) = the center and r = the radius

solution is

(x-2)^2 + y^2 = 50

plug in each of the 3 points to generate 3 equations, for 3 unknowns. Solve by substitution, elimination or matrix algebra

it may help to plot the points and draw a rough sketch of a circle through them, to get a general idea where the center is and how long the radius is.

(7 - h)^2 + (5-k)^2 = r^2

(1-h)^2 + (-7-k)^2 = r^2

(9-h)^2 + (-1-k)^2 = r^2

set the 1st two equal

49 -14h + h^2 + 25 - 10k + k^2 = 1 -2h + h^2 + 49 +14k + k^2

set the last two equations equal

81-18h + h^2 + 1 + 2k + k^2 = 1 -2h +h^2 +49 +14k +k^2

now you have two equations two unknowns

another way to view this problem is

use the distance formula from (h,k) to each of the 3 points

and set those distances equal. Each distance is the radius of the circle.

another method is find the circumcenter

you can find a circumcenter calculator on line, just plug in the 3 points

(7,5), (1,-7) and (9,-1) and out pops (2,0) = the center of the circle through the 3 points

the equation of the circle is (x-2)^2 + y^2 =50

r^2 = 7^2 +1 = 50

r^2 = 1^2 +(-7)^2 = 50

r^2 = 7^2 +(-1)^2 = 50

distance from the center to each point = r= sqr50