Question

The circle $x^2+y^2-2x=0$ makes an intercept $AB$ on the line $y=x.$ The circle on $AB$ as diameters makes intercepts of lengths

• A. $1$ unit on x-axis
• B. $2$ units on x-axis
• C. $1$ unit on y-axis
• D. $\sqrt{2}$ unit on $y=x$

Answer 1

Answer: (A), (C), (D)

The correct options are
A $1$ unit on x-axis
C $1$ unit on y-axis
D $\sqrt{2}$ unit on $y=x$

Equation of a circle passing through the intersection of the line $y=x$
and the circle $x^2+y^2-2x=0$ is $x^2+y^2-2x+\lambda (y-x)=0$
Since $y=x$ is a diameter of this circle
its centre $(\frac{\lambda +2}{2},-\frac{\lambda }{2})$ lies on it
$\Rightarrow \lambda =-1$
Thus the required equation of the circle is $x^2+y^2-x-y=0$.
Hence intercept on X-axis $=$ intercept on Y-axis $=1$
and intercept on $y=x$ is$=$ length of $AB=\sqrt{2}$