The equation of the hyperbola whose foci are ` ( +- 5,0) ` and eccentricity 5/3 is

Options

A. `x^(2)/9-y^(2)/16=1`
B. `x^(2)/4-y^(2)/(16)=1`
C. `x^(2)/9+y^(2)/6=1`
D. `x^(2)/4-y^(2)/(16)=1`

Answer

To find the equation of the hyperbola whose foci are at (±5, 0) and eccentricity is \( \frac{5}{3} \), we can follow these steps:

### Step 1: Identify the center and foci
The foci of the hyperbola are given as (±5, 0). This indicates that the center of the hyperbola is at the origin (0, 0).

### Step 2: Determine the value of \( c \)
The distance from the center to each focus is denoted as \( c \). Since the foci are at (±5, 0), we have:
\[ c = 5 \]

### Step 3: Use the eccentricity to find \( a \)
The eccentricity \( e \) of a hyperbola is defined as:
\[ e = \frac{c}{a} \]
Given that \( e = \frac{5}{3} \) and \( c = 5 \), we can substitute these values into the equation:
\[ \frac{5}{3} = \frac{5}{a} \]
Cross-multiplying gives us:
\[ 5 = \frac{5a}{3} \]
Multiplying both sides by 3:
\[ 15 = 5a \]
Dividing both sides by 5:
\[ a = 3 \]

### Step 4: Find \( b \) using the relationship between \( a \), \( b \), and \( c \)
The relationship between \( a \), \( b \), and \( c \) in a hyperbola is given by:
\[ c^2 = a^2 + b^2 \]
Substituting the known values:
\[ 5^2 = 3^2 + b^2 \]
This simplifies to:
\[ 25 = 9 + b^2 \]
Subtracting 9 from both sides:
\[ b^2 = 16 \]
Taking the square root gives:
\[ b = 4 \]

### Step 5: Write the equation of the hyperbola
Since the hyperbola opens horizontally (as the foci are along the x-axis), the standard form of the equation is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Substituting the values of \( a \) and \( b \):
\[ \frac{x^2}{3^2} - \frac{y^2}{4^2} = 1 \]
This simplifies to:
\[ \frac{x^2}{9} - \frac{y^2}{16} = 1 \]

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

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