Animated Solution for Mathematics - Circles: 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 .........

We are tasked with finding the equation of a circle that uses the chord $AB$ as its diameter. The chord $AB$ is formed by the intersection of the circle $x^2+y^2-2x=0$ and the line $y=x$.

The Collision

To find the endpoints of our diameter, we must determine where the line and the circle intersect. Since the points $A$ and $B$ must satisfy both equations simultaneously, we substitute $y=x$ into the circle equation:

This simplifies to the quadratic equation:

Factoring this expression, we obtain $2x(x-1)=0$, which yields two solutions: $x=0$ and $x=1$.

Using the relation $y=x$, we find the corresponding coordinates: If $x=0$, then $y=0$, giving us point $A(0,0)$. If $x=1$, then $y=1$, giving us point $B(1,1)$.

The Shortcut

While one could calculate the center and radius of the new circle, we can utilize the diameter form of a circle equation. This is a highly efficient method for JEE problems.

The formula for a circle with diameter endpoints $(x_1,y_1)$ and $(x_2,y_2)$ is:

The Final Synthesis

Substituting our points $A(0,0)$ and $B(1,1)$ into the diameter form equation, we get:

Expanding this expression, we arrive at:

Rearranging the terms, we reach the final equation of the circle:

This is the equation of the circle that perfectly embraces the chord $AB$ as its diameter. The elegance of this result demonstrates how choosing the correct geometric property simplifies complex coordinate problems.