Q. 144.0( 7 Votes )

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 :

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



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