Key Concepts and Formulas

• Equation of a Circle (Diameter Form): If $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of a diameter of a circle, then the equation of the circle is $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$.
• Intersection of a Line and a Circle: To find the points where a line intersects a circle, substitute the equation of the line into the equation of the circle and solve for the coordinates.

Step-by-Step Solution

Step 1: Find the Intersection Points of the Line and the Circle

Goal: Determine the coordinates of points A and B where the line $y=x$ intersects the circle $x^2 + y^2 - 2x = 0$. These points define the diameter of the circle we want to find.

Given Information:

• Line: $y = x$
• Circle: $x^2 + y^2 - 2x = 0$

Procedure: Substitute the equation of the line, $y = x$, into the equation of the circle: $x^2 + (x)^2 - 2x = 0$ Simplify and solve for $x$: $2x^2 - 2x = 0$ Factor out $2x$: $2x(x - 1) = 0$ This gives us two possible values for $x$:

• $2x = 0 \implies x = 0$
• $x - 1 = 0 \implies x = 1$

Now, find the corresponding $y$-values using the line equation $y = x$:

• If $x = 0$, then $y = 0$. So, point A is $(0, 0)$.
• If $x = 1$, then $y = 1$. So, point B is $(1, 1)$.

Explanation: We solved the system of equations to find the points where the line and circle intersect. These intersection points are the endpoints of the diameter AB for the circle we are looking for.

Step 2: Apply the Diameter Form of the Circle Equation

Goal: Find the equation of the circle with diameter AB, where A is $(0, 0)$ and B is $(1, 1)$.

Procedure: Use the diameter form of the circle's equation: $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$. Substitute the coordinates of A and B: $(x - 0)(x - 1) + (y - 0)(y - 1) = 0$ Expand and simplify: $x(x - 1) + y(y - 1) = 0$ $x^2 - x + y^2 - y = 0$ Rearrange to the standard form: $x^2 + y^2 - x - y = 0$

Explanation: We used the diameter form to directly calculate the equation of the circle given the endpoints of its diameter. This is a straightforward application of the formula.

Common Mistakes & Tips

• Understanding Intercepts: The intercept is the segment of the line that lies within the circle. Its endpoints are the intersection points.
• Alternative Method (Center-Radius Form): You could find the center as the midpoint of AB, and the radius as half the length of AB. Then, use the standard equation of a circle $(x-h)^2 + (y-k)^2 = r^2$.
• Algebraic Errors: Pay close attention to signs and expansion during the substitution and simplification steps.

Summary

We found the intersection points of the line $y=x$ and the circle $x^2 + y^2 - 2x = 0$, which are the endpoints of the diameter of the circle we want to find. Then, we applied the diameter form of the circle equation to obtain the equation $x^2 + y^2 - x - y = 0$.

The final answer is \boxed{x^2 + y^2 - x - y = 0}, which corresponds to option (A).