Question 175

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

Solution

The given circle is $$x^2+y^2-2x=0.$$

Substituting $$y=x,$$

we get

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

$$2x^2-2x=0.$$

$$2x(x-1)=0.$$

Hence the points of intersection are $$A(0,0)$$ and $$B(1,1).$$

The circle having $$AB$$ as diameter has equation $$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0.$$

Substituting

$$A(0,0),\qquad B(1,1),$$

$$x(x-1)+y(y-1)=0.$$

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

Therefore, the required circle is $$\boxed{x^2+y^2-x-y=0}.$$