Question

Find the equation of the circle which passes through the points A(1, 1) and B(2, 2) and whose radius is 1. Show that there are two such circles.

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 A(1, 1) in (i) 

(1 - h)² + (1 - k)² = 1²

⟹h² + k² + 2- 2h - 2k = 1

⟹h² + k² - 2h - 2k = 1 ....(ii)

Putting B(2,2) in (i)

(2-h)² + (2-k)² = 1²

⟹h² + k² + 8 - 4h - 4k = 1

h² + k² - 4h - 4k = -7

(h² + k² - 2h - 2k) - 2h - 2k = -7

⟹ - 1 - 2h - 2k = -7 [from (ii)]

⟹-2h - 2k = -6

⟹ h+ k = 3

⟹ h = 3 - k

Putting it in (ii)

⟹ (3 - k)² + k² - 2(3 - k) - 2k = -1

⟹ 9 + 2k² - 6k - 6 + 2k - 2k = -1

⟹ 2k² + 4 - 6k = 0

⟹k² - 3k + 2 = 0

⟹ k = 2 or k = 1

When k = 2, h = 3 - 2 = 1 

Equation of 1 circle 

(x - 1)² + (y - 2)² = 1 

When k = 1, h = 3 - 1 = 2

(x - 2)² + (y - 1)² = 1