Question

The intercept on the line $ y = x$ by the circle $ x^{2} + y^{2} - 2x = 0$ is $AB$. The 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} - 3 x + y = 0 $
• D. $x^{2} + y^{2} + 3x - y = 0 $

Answer 1

Answer: (B)

We have equation of line $y=x$ and equation of circle $x^{2}+y^{2}-2 x=0$

Now, intersecting points of given line and circle,
$ x^{2}+x^{2}-2 x =0$
$\Rightarrow 2 x^{2}-2 x =0$
$ \Rightarrow 2 x(x-1) =0 $
$ \Rightarrow x =0,1 $
when $x=0$ then $y=0$ and when $x=1$ then $y=1$
$\therefore $ Coordinates of end points of diameter $A B$ are (0,0) and (1,1)
$\therefore $ Required equation of circle with diameter $A B$
$(x-0)(x-1)+(y-0)(y-1)=0$
$\Rightarrow x^{2}-2+y^{2}-y=0 $
$\Rightarrow x^{2}+y^{2}-x-y=0$