If the distance between the foci of a hyperbola with x-axis as the major axis is 16 units and its eccentricity is `(4)/(3)`, then its equation is

Options

A. `7x^(2) - y^(2) =252`
B. `7x^(2) -9y^(2) =252`
C. `7x^(2) -y^(2) =252`
D. `9x^(2) - y^(2) =252`

Answer

To solve the problem step-by-step, let's follow the reasoning laid out in the video transcript.

### Step 1: Understanding the Given Information
We know that:
- The distance between the foci of the hyperbola is 16 units.
- The eccentricity (E) of the hyperbola is \( \frac{4}{3} \).

### Step 2: Relating the Distance Between Foci to a and E
The distance between the foci of a hyperbola is given by the formula:
\[ \text{Distance between foci} = 2ae \]
Given that this distance is 16, we can set up the equation:
\[ 2ae = 16 \]

### Step 3: Solving for a
Substituting the value of eccentricity (E) into the equation:
\[ 2a \left( \frac{4}{3} \right) = 16 \]
Now, simplify this equation:
\[ \frac{8a}{3} = 16 \]
To isolate \( a \), multiply both sides by \( \frac{3}{8} \):
\[ a = 16 \cdot \frac{3}{8} = 6 \]

### Step 4: Finding b using the Relationship with E
We know that for a hyperbola:
\[ b^2 = a^2(e^2 - 1) \]
First, we need to calculate \( e^2 \):
\[ e^2 = \left( \frac{4}{3} \right)^2 = \frac{16}{9} \]
Now, substitute \( a = 6 \) into the equation:
\[ b^2 = 6^2 \left( \frac{16}{9} - 1 \right) \]
Calculating \( \frac{16}{9} - 1 \):
\[ \frac{16}{9} - 1 = \frac{16}{9} - \frac{9}{9} = \frac{7}{9} \]
Now substitute back into the equation for \( b^2 \):
\[ b^2 = 36 \cdot \frac{7}{9} = 28 \]

### Step 5: Writing the Standard Form of the Hyperbola
The standard form of the equation of a hyperbola with the x-axis as the transverse axis is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Substituting \( a^2 = 36 \) and \( b^2 = 28 \):
\[ \frac{x^2}{36} - \frac{y^2}{28} = 1 \]

### Step 6: Simplifying the Equation
To express this in a more standard form, we can multiply through by the least common multiple of 36 and 28, which is 252:
\[ 252 \left( \frac{x^2}{36} - \frac{y^2}{28} \right) = 252 \]
This simplifies to:
\[ 7x^2 - 9y^2 = 252 \]

### Final Equation
Thus, the equation of the hyperbola is:
\[ 7x^2 - 9y^2 = 252 \]