Find the standard equation of the hyperbola which satisﬁes the given condition:

asymptotes 

y=43x+13 and y=−43x+413,a vertex (−1,7)


To find the standard equation of the hyperbola that satisfies the given conditions, we need to analyze the information provided:

1. **Asymptotes:** The given asymptotes are:
   - \( y = 43x + 13 \)
   - \( y = -43x + 413 \)

2. **Vertex:** A vertex of the hyperbola is given at \( (-1, 7) \).

### Step 1: Determine the Slope of the Asymptotes

For a hyperbola centered at \((h, k)\) with horizontal transverse axis, the equations of the asymptotes are:

\[
y - k = \frac{b}{a}(x - h) \quad \text{and} \quad y - k = -\frac{b}{a}(x - h)
\]

From the given equations, the slopes of the asymptotes are \( m_1 = 43 \) and \( m_2 = -43 \).

Thus, we can write:

\[
\frac{b}{a} = 43
\]

### Step 2: Find the Center \((h, k)\)

To find the center \((h, k)\) of the hyperbola, we need the midpoint of the y-intercepts of the asymptotes. The y-intercepts of the given asymptotes are:

- For \( y = 43x + 13 \), y-intercept is 13.
- For \( y = -43x + 413 \), y-intercept is 413.

The midpoint of these y-intercepts is:

\[
k = \frac{13 + 413}{2} = \frac{426}{2} = 213
\]

The x-coordinates of the asymptotes are the same, which means the x-coordinate of the center \(h\) is the same as that of the vertex, which is \(-1\). 

So, the center is \((-1, 213)\).

### Step 3: Write the Equation of the Hyperbola

For a hyperbola centered at \((h, k) = (-1, 213)\) with a horizontal transverse axis, the standard form is:

\[
\frac{(x + 1)^2}{a^2} - \frac{(y - 213)^2}{b^2} = 1
\]

From the slopes of the asymptotes, we have:

\[
\frac{b}{a} = 43 \implies b = 43a
\]

### Step 4: Use the Vertex to Find \(a^2\)

The distance from the center to a vertex is \(a\). Since the vertex is at \((-1, 7)\), the distance between the center \((-1, 213)\) and the vertex \((-1, 7)\) is:

\[
|213 - 7| = 206
\]

Therefore, \(a = 206\). 

Thus, \(a^2 = 206^2 = 42436\).

### Step 5: Determine \(b^2\)

Since \(b = 43a\), we have:

\[
b = 43 \times 206 = 8858
\]

Therefore, 

\[
b^2 = 8858^2 = 78461464
\]

### Step 6: Write the Final Equation

Substitute \(a^2\) and \(b^2\) into the standard form:

\[
\frac{(x + 1)^2}{42436} - \frac{(y - 213)^2}{78461464} = 1
\]

### Conclusion

The standard equation of the hyperbola is:

\[
\frac{(x + 1)^2}{42436} - \frac{(y - 213)^2}{78461464} = 1
\]

Would you like more details on any step, or do you have any questions?

### Related Questions:

1. How do you determine the equation of an ellipse given its foci and vertices?
2. What is the difference between the equations of hyperbolas with horizontal and vertical transverse axes?
3. How do you find the asymptotes of a hyperbola given its equation?
4. What are the real-world applications of hyperbolas?
5. How can we derive the formula for the eccentricity of a hyperbola?

### Tip:  
When solving problems involving conic sections, always pay attention to the orientation (horizontal or vertical) and the location of the center.

Learn how to find the standard equation of a hyperbola with given asymptotes y = 43x + 13 and y = -43x + 413, and a vertex at (-1, 7). This problem involves analyzing asymptotes, determining the center of the hyperbola, calculating distances, and writing the standard form equation. Suitable for advanced high school and college students.

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