Animated Solution for Mathematics - Circles: 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

Imagine you are standing on the Cartesian plane, looking at a circle defined by the equation:

A line defined by $y=x$ slices through this circle, creating a chord $AB$ that connects the two points of intersection. Our objective is to determine the equation of a new circle that uses this segment $AB$ as its diameter.

To build our new circle, we must first identify the coordinates of points $A$ and $B$. We solve the equations of the line and the circle simultaneously.

Substituting $y=x$ into the circle's equation $x^2+y^2-2x=0$, we obtain:

Simplifying this expression yields:

Factoring the quadratic gives $2x(x-1)=0$, which indicates that the intersection occurs at $x=0$ and $x=1$. Since the line is defined by $y=x$, the corresponding $y$ coordinates are also $0$ and $1$.

Thus, our anchor points are $A(0,0)$ and $B(1,1)$.

While we could calculate the midpoint for the center and the distance for the radius, we can utilize the more elegant diameter form of a circle equation.

If a circle has a diameter with endpoints $(x_1,y_1)$ and $(x_2,y_2)$, its equation is given by:

This formula is derived from the geometric property that any point $(x,y)$ on the circle forms a right angle with the endpoints of the diameter. Substituting our points $A(0,0)$ and $B(1,1)$ into this formula, we get:

Expanding the expression above, we obtain:

Rearranging the terms into the standard form, we arrive at the final equation for our new circle: