https://file.aiquickdraw.com/mathbot/file/1728502772477s0oxvzdm.jpg

The image presents a problem involving a line and a circle. The problem can be interpreted as follows:

- The equation of the line is given by \( y = 2x - 2 \).
- The equation of the circle is \( (x - 2)^2 + (y - 2)^2 = 20 \).

The tasks are:

**a.** Find the coordinates of the points \( A \) and \( B \) where the line intersects the circle.

**b.** Show that \( AB \) is a diameter of the circle.

### Solution (a):

We need to find the points where the line \( y = 2x - 2 \) intersects the circle. To do this, substitute the equation of the line into the equation of the circle.

1. Substituting \( y = 2x - 2 \) into \( (x - 2)^2 + (y - 2)^2 = 20 \):

   \[
   (x - 2)^2 + ((2x - 2) - 2)^2 = 20
   \]
   
2. Simplify the equation:
   \[
   (x - 2)^2 + (2x - 4)^2 = 20
   \]
   Expanding both terms:
   \[
   (x - 2)^2 = x^2 - 4x + 4
   \]
   \[
   (2x - 4)^2 = 4x^2 - 16x + 16
   \]

3. Substituting into the equation:
   \[
   x^2 - 4x + 4 + 4x^2 - 16x + 16 = 20
   \]

4. Combine like terms:
   \[
   5x^2 - 20x + 20 = 20
   \]

5. Simplify:
   \[
   5x^2 - 20x = 0
   \]

6. Factor:
   \[
   5x(x - 4) = 0
   \]

7. Solve for \( x \):
   \[
   x = 0 \quad \text{or} \quad x = 4
   \]

8. Now, substitute these values of \( x \) back into the equation \( y = 2x - 2 \) to find the corresponding \( y \)-coordinates:

   - For \( x = 0 \): 
     \[
     y = 2(0) - 2 = -2
     \]
     So, \( A(0, -2) \).

   - For \( x = 4 \): 
     \[
     y = 2(4) - 2 = 6
     \]
     So, \( B(4, 6) \).

Thus, the coordinates of the points of intersection are \( A(0, -2) \) and \( B(4, 6) \).

### Solution (b):

To show that \( AB \) is a diameter, we need to check if the distance between \( A \) and \( B \) is equal to the diameter of the circle.

1. The distance between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the distance formula:
   \[
   AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]

   Substituting the coordinates of \( A(0, -2) \) and \( B(4, 6) \):

   \[
   AB = \sqrt{(4 - 0)^2 + (6 - (-2))^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}
   \]

2. The diameter of the circle is twice the radius. From the equation of the circle \( (x - 2)^2 + (y - 2)^2 = 20 \), we know that the radius is \( \sqrt{20} = 2\sqrt{5} \).

3. Therefore, the diameter is \( 2 \times 2\sqrt{5} = 4\sqrt{5} \), which is exactly the length of \( AB \).

Since the length of \( AB \) is equal to the diameter of the circle, we have shown that \( AB \) is a diameter of the circle.

---

Do you have any further questions or need additional details on this problem?

### Related questions:
1. What is the general formula for finding points of intersection between a line and a circle?
2. How can you derive the equation of a circle from its geometric properties?
3. How does the discriminant of a quadratic equation relate to the number of intersection points between a line and a circle?
4. Can you explain why a diameter always passes through the center of a circle?
5. What is the significance of a line intersecting a circle at two distinct points?

### Tip:
For problems involving both a line and a circle, always try substituting the linear equation into the circular equation to find points of intersection. This reduces the problem to solving a quadratic equation, making it easier to interpret the geometry involved.

The line y = 2x - 2 meets the circle (x - 2)^2 + (y - 2)^2 = 20 at A and B. a. Find the coordinates of A and B. b. Show that AB is a diameter of the circle.

Learn how to find the points of intersection between a line and a circle, and verify whether the line segment is a diameter. This problem explores algebraic methods and geometric properties, including solving quadratic equations and applying the distance formula.

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