To determine the coordinates of point $B$, where the line intersects the circle, we need to analyze the geometric setup and find the equation of the line passing through points $A$ and $C$, as well as the equation of the circle.

Step 1: Equation of the Circle

The circle is centered at $(0, r)$ with a radius $r$. The equation of a circle centered at $(h, k)$ with radius $r$ is: $(x - h)^2 + (y - k)^2 = r^2$ Here, $h = 0$ and $k = r$, so the equation of the circle becomes: $x^2 + (y - r)^2 = r^2$ Expanding this gives: $x^2 + y^2 - 2ry + r^2 = r^2$ Simplifying: $x^2 + y^2 - 2ry = 0$ Thus, the equation of the circle is: $x^2 + y^2 - 2ry = 0$

Step 2: Equation of the Line through Points $A$ and $C$

The line passes through points $A(0, 2r)$ and $C(9r^2, 0)$. The slope $m$ of the line is: $m = \frac{0 - 2r}{9r^2 - 0} = \frac{-2r}{9r^2} = \frac{-2}{9r}$ Thus, the equation of the line in point-slope form is: $y - 2r = \frac{-2}{9r}(x - 0)$ Simplifying: $y = \frac{-2}{9r}x + 2r$

Step 3: Solve for Intersection of the Line and Circle

We now substitute the equation of the line $y = \frac{-2}{9r}x + 2r$ into the equation of the circle $x^2 + y^2 - 2ry = 0$.

1. Substitute $y = \frac{-2}{9r}x + 2r$ into the circle equation: $x^2 + \left(\frac{-2}{9r}x + 2r\right)^2 - 2r\left(\frac{-2}{9r}x + 2r\right) = 0$

2. Expand the terms carefully to simplify the equation.

This will give a quadratic equation in $x$, whose solution will yield the $x$-coordinate of point $B$. Then, substitute that $x$-value back into the line equation to find the $y$-coordinate.

Would you like me to fully expand and solve this system, or focus on a specific part of the solution process? Feel free to ask any questions!

Here are 5 follow-up questions to expand on this topic:

1. How can we use the discriminant to check how many intersection points exist between the line and the circle?
2. How does the value of $r$ affect the position of point $B$?
3. Can you explain how to interpret the slope of the line geometrically?
4. What happens to the coordinates of $B$ as $r \to \infty$?
5. How do we derive the equation of a tangent line to a circle at a given point?

Tip: When dealing with conic sections like circles and lines, visualizing the geometric relationships can greatly aid in setting up the correct equations for intersections!