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.