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The intercept on the line y=x by the cir...

The intercept on the line `y=x` by the circle `x^2+y^2-2x=0` is AB. Equation of the circle with AB as a diameter is

A

`x^2+y^2+x+y=0`

B

`x^2+y^2-x-y=0`

C

`x^2+y^2+x-y=0`

D

`x^(2)+y^(2)-x+y=0`

Text Solution

Verified by Experts

The correct Answer is:
`x^(2)+y^(2)-x-y=0`
Equation of any circle passing through the point of lt brgt intersection of `x^(2)+y^(2)-2x=0 and y=x`is
`(x^(2)+y^(2)-2x)+lambda(y-x)=0`
`rArr x^(2)+y^(2)-(2+lambda)x+lambday=0`
Its centre is `((2+lambda)/(2),(-lambda)/(2))`.
`rArr` For AB to be the diameter of the required circle the centre must be on AB, i.e.
`2+lambda=-lambda`
`rArr lambda = -1 " "[therefore y = x]`
therefore, equation of the required circle is
`x^(2)+y^(2)=(2-1)x-1*y=0`
`rArr x^(2)+y^(2)-x-y=0`
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