Key Concepts and Formulas

• The equation of a circle with endpoints of a diameter at $(x_1, y_1)$ and $(x_2, y_2)$ is given by $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$.
• If a point lies on a circle, its coordinates satisfy the equation of the circle.

Step-by-Step Solution

Step 1: Identify the Diameter

We are given the points $(2k, 3k)$, $(1,0)$, $(0,1)$, and $(0,0)$. Since the points $(1,0)$, $(0,1)$, and $(0,0)$ lie on the circle, we can observe that the lines connecting $(0,0)$ to $(1,0)$ and $(0,0)$ to $(0,1)$ are perpendicular (they lie on the x and y axes respectively). Therefore, the points $(1,0)$ and $(0,1)$ define a diameter of the circle.

Step 2: Form the Equation of the Circle

Using the diameter form of the circle equation with endpoints $(1,0)$ and $(0,1)$, we have: $(x - 1)(x - 0) + (y - 0)(y - 1) = 0$ $x(x - 1) + y(y - 1) = 0$ $x^2 - x + y^2 - y = 0$ $x^2 + y^2 - x - y = 0$

This is the equation of the circle passing through $(1,0)$, $(0,1)$, and $(0,0)$.

Step 3: Substitute the Fourth Point into the Circle Equation

Since the point $(2k, 3k)$ also lies on the circle, it must satisfy the equation we derived. Substituting $x = 2k$ and $y = 3k$ into the equation $x^2 + y^2 - x - y = 0$, we get: $(2k)^2 + (3k)^2 - (2k) - (3k) = 0$ $4k^2 + 9k^2 - 2k - 3k = 0$ $13k^2 - 5k = 0$

Step 4: Solve for k

We can factor out a $k$ from the equation: $k(13k - 5) = 0$ This gives us two possible solutions for $k$: $k = 0$ or $13k - 5 = 0$. Since the points must be distinct, $k$ cannot be $0$. Therefore, we have: $13k - 5 = 0$ $13k = 5$ $k = \frac{5}{13}$

Common Mistakes & Tips

• Remember that the diameter form of the circle equation is a powerful tool when you know the endpoints of a diameter.
• Always check if the solutions you obtain make sense in the context of the problem. In this case, $k=0$ would make all four points not distinct, so we disregard it.
• Be careful with algebraic manipulations and signs when substituting and simplifying.

Summary

We used the fact that the points $(1,0)$ and $(0,1)$ form a diameter of the circle to find the equation of the circle. Then, we substituted the coordinates of the fourth point $(2k, 3k)$ into the equation and solved for $k$. We found that $k = \frac{5}{13}$.

Final Answer The final answer is $\boxed{\frac{5}{13}}$, which corresponds to option (C).