Find the equation of the straight line which is parallel to the x axis and crosses the circle x2 + y2 = 9 in points A and B in such a way that |AB| = 2. Consider all possible solutions.

We are tasked with finding the equation of the straight line that is parallel to the x-axis and intersects the circle \( x^2 + y^2 = 9 \) in points \( A \) and \( B \), such that \( |AB| = 2 \).

### Step 1: Analyze the equation of the circle

The given equation of the circle is:

\[
x^2 + y^2 = 9
\]

This is the equation of a circle centered at \( (0, 0) \) with a radius of \( 3 \).

### Step 2: Equation of the line parallel to the x-axis

A line parallel to the x-axis has an equation of the form:

\[
y = c
\]

where \( c \) is a constant representing the y-coordinate of all points on the line.

### Step 3: Points of intersection of the line and the circle

To find the points of intersection of the line \( y = c \) with the circle \( x^2 + y^2 = 9 \), substitute \( y = c \) into the equation of the circle:

\[
x^2 + c^2 = 9
\]

Solving for \( x \):

\[
x^2 = 9 - c^2
\]

\[
x = \pm \sqrt{9 - c^2}
\]

Thus, the points of intersection are \( A(-\sqrt{9 - c^2}, c) \) and \( B(\sqrt{9 - c^2}, c) \).

### Step 4: Distance between points \( A \) and \( B \)

The distance between \( A(-\sqrt{9 - c^2}, c) \) and \( B(\sqrt{9 - c^2}, c) \) is the horizontal distance, as they have the same y-coordinate \( c \):

\[
|AB| = 2\sqrt{9 - c^2}
\]

We are given that \( |AB| = 2 \), so:

\[
2\sqrt{9 - c^2} = 2
\]

### Step 5: Solve for \( c \)

Divide both sides by 2:

\[
\sqrt{9 - c^2} = 1
\]

Square both sides:

\[
9 - c^2 = 1
\]

\[
c^2 = 8
\]

\[
c = \pm \sqrt{8} = \pm 2\sqrt{2}
\]

### Step 6: Equation of the line

Thus, the equation of the line can be:

\[
y = 2\sqrt{2} \quad \text{or} \quad y = -2\sqrt{2}
\]

These are the two possible solutions.

### Final Answer:
The equations of the straight line are:

\[
y = 2\sqrt{2} \quad \text{and} \quad y = -2\sqrt{2}
\]

Would you like more details or clarifications?

Here are 5 related questions:
1. How do we derive the equation of a circle from its geometric properties?
2. What is the significance of the radius in the equation \(x^2 + y^2 = r^2\)?
3. How can we find the distance between two points on a coordinate plane?
4. Why does a line parallel to the x-axis have a constant y-value?
5. Can you find the points \( A \) and \( B \) for each solution?

**Tip**: For lines parallel to the x-axis, the y-coordinate remains constant, simplifying the geometry when intersecting with circles.

Find the equation of the straight line which is parallel to the x axis and crosses the circle x^2 + y^2 = 9 in points A and B in such a way that |AB| = 2. Consider all possible solutions.

Discover how to find the equation of a line parallel to the x-axis that crosses the circle x^2 + y^2 = 9 at points A and B, where the distance |AB| equals 2. This problem involves key concepts in geometry, including circles and distance formulas. Suitable for high school students in Grades 10-12.

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