Derive the equation of a hyperbola centered at (0,0) with its transverse axis along the x-axis.

The general equation of a hyperbola centered at (0,0) with its transverse axis along the x-axis is:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

where a is the distance from the center to the vertices along the x-axis (transverse axis) and b is the distance from the center to the vertices along the y-axis (conjugate axis).

Since the hyperbola is centered at (0,0) and its transverse axis is along the x-axis, the equation simplifies to:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

where a is the distance from the center to the vertices along the x-axis.

Therefore, the equation of the hyperbola centered at (0,0) with its transverse axis along the x-axis is:

\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

Step 1: Define the Coordinates

Let (x, y) be any point on the hyperbola.

Step 2: Transverse Axis and Semi-Transverse Axis

Since the transverse axis is along the x-axis, the distance from the center (0,0) to a point on the hyperbola along the x-axis is:

```
a = distance from center to vertex
```

Step 3: Conjugate Axis and Semi-Conjugate Axis

The conjugate axis is perpendicular to the transverse axis and passes through the center. The distance from the center to a point on the hyperbola along the y-axis is:

```
b = distance from center to co-vertex
```

Step 4: Focal Points

The foci of the hyperbola lie on the transverse axis, a distance c from the center:

```
c = distance from center to foci
```

Step 5: Relationship between a, b, and c

For a hyperbola, the following relationship holds:

```
c² = a² + b²
```

Step 6: Rectangle Properties

Draw a rectangle with vertices at (±a, ±b). The diagonals of the rectangle intersect at the center (0,0).

The length of the diagonals is:

```
2a√2
```

Step 7: Equation of the Hyperbola

The equation of a hyperbola centered at (0,0) with its transverse axis along the x-axis is given by:

```
(x²/a²) - (y²/b²) = 1
```

Proof:

Let (x, y) be any point on the hyperbola. Then, the distance from (x, y) to (±a, 0) is:

```
√[(x - a)² + y²]
```

Similarly, the distance from (x, y) to (±a, 0) is:

```
√[(x + a)² + y²]
```

Since the diagonals of the rectangle intersect at the center, we have:

```
2√[(x - a)² + y²] = 2√[(x + a)² + y²]
```

Simplifying, we get:

```
x² - a² = y² - b²
```

Dividing by a²b², we obtain:

```
(x²/a²) - (y²/b²) = 1
```

Therefore, the equation of the hyperbola centered at (0,0) with its transverse axis along the x-axis is:

```
(x²/a²) - (y²/b²) = 1
```