Question

A(-2, -3), B(-1, 0) and C(7, -6) are the vertices of a triangle. Find the circumcentre and the circumradius of the triangle. 

Answer 1

Circumcircle of MBCwill pass through the vertices A,B and C. AB = OB  (radii of same circle)

⇒ OA² =OB² 

(x+2)² +(y+3)²=(x+1)² +Y² 

⇒  4x + 4 + 9 + 6y = 2x + 1 

⇒  2x+6y=-12  .....(1)

OB =OC (radii of same circle) 

⇒ OB² =OC² 

(x + 1)² + y² =(x - 7)² +(Y+ 6)²  

⇒ 2x + 1= 49 - 144 x + 36 + 12 

⇒  16x - 12=8y 

⇒ 4x - 3y = 21  ....(2)

On solving (1) and (2)

4x + 12y = -24

4x  -  3y   = 21
-      +      -
_____________  

15 y = - 45

y = - 3

from (1)

2x + 6(-3) = -12

⇒ 2x = 6  

⇒ x = 3

Circumcentre of  Δ ABC is (3, - 3)and 

Circumradius = `sqrt ((3 + 2)^2 + (-3 + 3)^2) = sqrt 25 = 5  "units"`