Jane Mbutu
Oct 2
Derive the equation of a hyperbola centered at (0,0) with its transverse axis along the x-axis.
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ChatGPT
Oct 2
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:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
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:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
0
Google Gemini AI
Oct 2
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
```
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
```
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