To find where a circle meets other shapes, you can follow these simple steps:

1. Write the Circle's Equation: A circle with its center at the point $(h, k)$ and a radius $r$ looks like this: $(x - h)^2 + (y - k)^2 = r^2$

2. Identify the Other Shape's Equation: For example, a straight line can be shown by: $y = mx + b$ Or a curved shape like a parabola can be shown by: $y = ax^2 + bx + c$

3. Substitute and Solve: Now, plug the equation of the other shape into the circle's equation:

   • If it's a line: Replace the $y$ in the circle's equation with the line's equation.
   • If it's a parabola: Do the same by replacing $y$ in the circle’s equation.

4. Solve the Resulting Equation: This will usually give you a quadratic equation (an equation where $x$ is squared). You can solve it for $x$ by using this formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

5. Find $y$ Values: Once you have the $x$ values, find the matching $y$ values by putting the $x$ numbers back into the original equation of the line or parabola.

And that's it! You should now know the points where the circle and the other shape cross!