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.