Show that the following four points in each of the following are con cyclic and find the equation of the - Maths - Relations and Functions

Question

Show that the following four points in each of the following are
con cyclic and find the equation of the circle
(1,1),(-6,0),(-2,2),(-2,-8)

Avinash Garg · Accepted Answer

Dear Student

the general equation of a circle is x2+y2+2gx+2fy+c=0----(1)point (1,1) lies on (1);1+1+2g+2f+c=0⇒2g+2f+c+2=0------(2)also point (-6,0) lies on (1)so 36+0-12g+c=0⇒36-12g+c=0--------(3)point (-2,2) lies on (1)so 4+4-4g+4f+c=0⇒8-4g+4f+c=0---------(4)point (-2,-8) lies on (1)so 4+64-4g-16f+c=0⇒68-4g-16f+c=0--------(5)on solving (2) and (3) for elemination of c we get14g+2f-34=0-------(6)on solving (4) and (5) for elemination of c we get20f-60=0⇒f=3; put this value in (6)14g+6-34=0⇒g=2; put this value in (3) we get36-12×2+c=0c=-12put the values of f,g and c in (1); we get x2+y2-4x+6y-12=0hence this is required equation of circle
Regards