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The given problem is to find the points of intersection between the circle and the line. The equations provided are:

1. **Circle Equation:** \((x - 3)^2 + (y - 1)^2 = 25\)
2. **Line Equation:** \(y = 2x\)

### Steps to find the intersection points:

1. **Substitute the line equation into the circle equation:**
   Since \( y = 2x \), we substitute \( y = 2x \) into the circle equation:

   \[
   (x - 3)^2 + (2x - 1)^2 = 25
   \]

2. **Expand and simplify:**
   \[
   (x - 3)^2 = x^2 - 6x + 9
   \]
   \[
   (2x - 1)^2 = 4x^2 - 4x + 1
   \]
   Adding these, we get:
   \[
   x^2 - 6x + 9 + 4x^2 - 4x + 1 = 25
   \]
   \[
   5x^2 - 10x + 10 = 25
   \]

3. **Form the quadratic equation:**
   \[
   5x^2 - 10x - 15 = 0
   \]
   Divide by 5:
   \[
   x^2 - 2x - 3 = 0
   \]
   Factorizing:
   \[
   (x - 3)(x + 1) = 0
   \]
   So, \( x = 3 \) and \( x = -1 \).

4. **Find corresponding \( y \)-coordinates:**
   For \( x = 3 \):
   \[
   y = 2(3) = 6
   \]
   So, one point is \( (3, 6) \).

   For \( x = -1 \):
   \[
   y = 2(-1) = -2
   \]
   So, the other point is \( (-1, -2) \).

### Points of Intersection:
The points of intersection are:
1. \((3, 6)\)
2. \((-1, -2)\)

Would you like a more detailed explanation of any of the steps? Here are some related questions:

1. How can you verify the intersection points?
2. What is the geometric interpretation of these solutions?
3. Can this method be generalized to any circle and line?
4. How would the solution change if the line was tangent to the circle?
5. What are the possible number of intersection points between a line and a circle?

**Tip:** For any geometric problem, it's useful to sketch the graph to visualize the solutions better!

Determine the coordinates of any points of intersection between the circle (x - 3)^2 + (y - 1)^2 = 25 and the line y = 2x.

Learn how to find the intersection points between the circle (x - 3)^2 + (y - 1)^2 = 25 and the line y = 2x. This problem involves solving a system of equations, substituting the line equation into the circle equation, and solving the resulting quadratic equation. Suitable for students in Grades 10-12.

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