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Let S1 be a circle passing through A(0,1...

Let `S_1` be a circle passing through `A(0,1)` and `B(-2,2)` and `S_2` be a circle of radius `sqrt(10)` units such that `A B` is the common chord of `S_1a n dS_2dot` Find the equation of `S_2dot`

Text Solution

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The correct Answer is:
`x^(2)+y^(2)+2x-3y+2+-sqrt(7)(x+2y-2)=0`
The equation of line AB is
`y-2=(2-1)/(-2-0)(x+2)=-(1)/(2)(x+2)`
or `x+2y-2=0` (1)
The equation of the circle whose diagonally opposite points are A and B is
`(x-0)(x+2)+(y-1)(y-2)=0`
or `x^(2)+Y^(2)+2x-3y+2=0` (2)
The family of circles passing through the points of intersection of (1) and (2) is
`x^(2)+y^(2)+2x-3y+2+lambda(x+2y-2)=0`
or `x^(2)+y^(2)+(2+lambda)x+(2lambda-3)y+2-2 lambda=0` (3)
Equation (3) represents a circle of radius `sqrt(10)` units . Therefore,
`sqrt((-(2+lambda)/(2))+(-(2lambda-3)/(2))^(2)-2+2lambda)=sqrt(10)`
or `(4+4lambda+lambda^(2))+(4lambda^(2)+9-12lambda)+8lambda-8=40`
or `lambda = +- sqrt(7)`
Hence, the required circles are
`x^(2)+y^(2)+2x-3y+2+-sqrt(7)(x+2y-2)=0`
There are two such circles possible.
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