Question

Show that the points A(1, 0), B(2, - 7), c(8, 1) and D(9, - 6) all lie on the same circle. Find the equation of this circle, its centre and radius.

Answer 1

The general equation of a circle: 

(x - h)² + (y - k)² = r² …(i), where (h, k) is the centre and r is the radius. 

Putting (1, 0) in (i) 

(1 - h)² + (0 - k)² = r² 

⇒h² + k² + 1 - 2h = r² ..(ii) 

Putting (2, - 7) in (i) 

(2 - h)² + ( - 7 - k)² = r²

 ⇒h² + k² + 53 - 4h + 14k = r² 

⇒ (h² + k² + 1 - 2h) + 52 - 2h + 14k = r² 

h - 7k - 26 = 0..(iii) [from (ii)] 

Similarly putting (8, 1) 

7h + k - 32 = 0..(iv)

Solving (iii)&(iv) 

h = 5 and k = - 3 

centre(5, - 3)

Radius = 25 

To check if (9, - 6) lies on the circle, (9 - 5)² + ( - 6 + 3)² = 5² 

Hence, proved.