Question

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. $Noneofthese$

Answer 1

Answer: (B)

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

Equation of any circle passing through the point of intersection $x^2+y^2-2x=0$ and $y=x$ is

$x^2+y^2-2x+\lambda (y-x)=0\Rightarrow x^2+y^2-(2+\lambda )x+\lambda y=0$

Its center is $(\frac{2+\lambda }{2},\frac{-\lambda }{2})$

For $AB$ to be the diameter of the required circle, the center must lie on $AB$ i.e.

$\frac{2+\lambda }{2}=\frac{-\lambda }{2}\Rightarrow \lambda =1$

Thus, equation of required circle is

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