Obtain the equation of a hyperbola with coordinate axes as principal axes given that the distances of one of its vertices from the foci are 9 and 1 units.

Answer

To solve the problem of finding the equation of a hyperbola with the coordinate axes as its principal axes, given that the distances of one of its vertices from the foci are 9 and 1 units, we can follow these steps:

### Step 1: Understand the properties of the hyperbola
The standard form of the equation of a hyperbola with its transverse axis along the x-axis is given by:

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

where:
- \( a \) is the distance from the center to each vertex,
- \( c \) is the distance from the center to each focus,
- \( e \) is the eccentricity, which is defined as \( e = \frac{c}{a} \).

### Step 2: Set up the relationships
From the problem, we know:
- The distance from one vertex to the foci is given as 9 and 1 units.
- The distance from the focus to the vertex can be expressed as \( c - a = 1 \) and \( c + a = 9 \).

### Step 3: Write equations based on the distances
From the distances we have:
1. \( c - a = 1 \) (Equation 1)
2. \( c + a = 9 \) (Equation 2)

### Step 4: Solve the equations
To find \( a \) and \( c \), we can add and subtract the two equations.

Adding Equation 1 and Equation 2:
\[ (c - a) + (c + a) = 1 + 9 \]
\[ 2c = 10 \implies c = 5 \]

Now, substituting \( c = 5 \) into Equation 1:
\[ 5 - a = 1 \implies a = 4 \]

### Step 5: Find the value of \( b \)
We know the relationship between \( a \), \( b \), and \( c \) is given by:
\[ c^2 = a^2 + b^2 \]
Substituting the known values:
\[ 5^2 = 4^2 + b^2 \]
\[ 25 = 16 + b^2 \]
\[ b^2 = 25 - 16 = 9 \implies b = 3 \]

### Step 6: Write the equation of the hyperbola
Now that we have \( a \) and \( b \), we can substitute these values into the standard form of the hyperbola:
\[ \frac{x^2}{4^2} - \frac{y^2}{3^2} = 1 \]
This simplifies to:
\[ \frac{x^2}{16} - \frac{y^2}{9} = 1 \]

### Final Answer
The equation of the hyperbola is:
\[ \frac{x^2}{16} - \frac{y^2}{9} = 1 \]

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