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Find the equation of the circles which passes through the points ` A(1, 1) and B(2, 2 )` and whose radius is 1. Show that there are two such circles. |
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Answer» Let the centre of the circle be `C(h, k )`. Then, `|CA| = 1 and |CB| = 1 ` ` rArr ( h - 1 ) ^(2) + (k - 1 ) ^(2) = 1 ^(2) and ( h- 2 ) ^(2) + (k - 2 ) ^(2) = 1 ^(2)` `rArr h ^(2) + k ^(2) - 2h - 2k + 1 = 0 " "` ... (i) and ` h ^(2) + k ^(2) - 4h - 4k + 7 = 0 " " ` ... (ii) Subtacting (ii) from (i), we get ` 2h + 2k - 6 = 0 rArr h + k = 3 rArr k = ( 3- h ) ` Putting this value in (i), we get ` h ^(2) + ( 3- h ) ^(2) - 2h - 2 ( 3- h ) + 1 = 0 ` `rArr ( h - 2 ) (h - 1 ) = 0 rArr h = 2 or h = 1`. ` therefore k = 1 or k = 2 ` Hence, the centres are `(2, 1 ) and (1, 2)` |
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