Question

P(5 , -8) , Q (2 , -9) and R(2 , 1) are the vertices of a triangle. Find tyhe circumcentre and the circumradius of the triangle.

Answer 1

Circumcircle of Δ PQR will pass through its vertices P , Q and R.

OP = OQ

⇒ OP² = OQ²

(x - 5)² + (y + 8)² = (x - 2)² + (y + 9)²

⇒ 25 - 10x + 64 + 16y = 4 - 4x + 81 + 18 y

c - 6x - 2y + 4 =0

OQ = OR      ...(radii of square circle)

OQ² = OR²

(x - 2)² + (y + 9)² = (x - 2)² + (y - 1)²

⇒ 81 + 18 y = 1 - 2y

⇒ 20 y = - 80

y = -4       ......(2)

-6x + 8+4 = 0  ......[from (2)]

⇒ -6x = -12

⇒ x = 2

Circumcentre of Δ PQR is O (2 , -4)

Circumcentre = `sqrt ((2 - 5)^2 + (-4 + 8)^2)`

`= sqrt (9 + 16) = sqrt 25` = 5 units