The foci of a hyperbola are at (±8,0). Find the distance from the center of the hyperbola to one of its vertices if the endpoints of the conjugate axis are at (0,±6).

To solve this, we can use the standard form of the equation of a hyperbola centered at the origin. The equation of a hyperbola with a horizontal transverse axis is:

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

Where:
- \(a\) is the distance from the center to each vertex.
- \(b\) is the distance from the center to each endpoint of the conjugate axis.
- The distance from the center to each focus is \(c\).

### Step 1: Identify the given information
- The foci are at \((\pm 8, 0)\), which means \(c = 8\).
- The endpoints of the conjugate axis are at \((0, \pm 6)\), so \(b = 6\).

### Step 2: Use the relationship between \(a\), \(b\), and \(c\)
For a hyperbola, the relationship between these values is:

\[
c^2 = a^2 + b^2
\]

### Step 3: Plug in the known values
We know \(c = 8\) and \(b = 6\). Substituting these into the equation:

\[
8^2 = a^2 + 6^2
\]

\[
64 = a^2 + 36
\]

### Step 4: Solve for \(a\)
\[
a^2 = 64 - 36
\]
\[
a^2 = 28
\]
\[
a = \sqrt{28} = 2\sqrt{7}
\]

Thus, the distance from the center of the hyperbola to one of its vertices is \(2\sqrt{7}\).

### Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How would the equation change if the transverse axis was vertical?
2. What is the full equation of the hyperbola based on this information?
3. How do the foci and vertices relate to the eccentricity of the hyperbola?
4. Can you derive the distance between the two foci in this case?
5. How do we find the asymptotes of this hyperbola?

**Tip**: In hyperbolas, the conjugate axis doesn't affect the location of the foci, but it does help define the shape of the hyperbola through the value of \(b\).

Solve a hyperbola problem by finding the distance from the center to a vertex when the foci are at (±8,0) and the endpoints of the conjugate axis are at (0,±6). Learn key formulas, including the relationship between a, b, and c, and the hyperbola equation. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation