Question

The foci of a hyperbola are (8, 3) and (0, 3) and eccentricity is \(\frac{4}{3}\). Then the length of the transverse axis is

• A. \(\frac{32}{3}\)
• B. 4
• C. 8
• D. \(\frac{8}{3}\)
• E. 6

Answer 1

Answer: (E)

Step 1: Identify the center of the hyperbola  
The foci of the hyperbola are given as \( (8,3) \) and \( (0,3) \). 
The center of the hyperbola is the midpoint of the foci: \[ \left( \frac{8+0}{2}, \frac{3+3}{2} \right) = \left( \frac{8}{2}, \frac{6}{2} \right) = (4,3) \]

Step 2: Find the values of \( c \) and \( a \) 
The distance of each focus from the center is: \[ c = \text{distance from } (4,3) \text{ to } (8,3) \] \[ c = |8 - 4| = 4 \] The eccentricity is given as: \[ e = \frac{4}{3} \] Using the formula for eccentricity of a hyperbola: \[ e = \frac{c}{a} \] Substituting values: \[ \frac{4}{3} = \frac{4}{a} \] Solving for \( a \): \[ a = \frac{4}{\frac{4}{3}} = \frac{4 \times 3}{4} = 3 \]

Step 3: Find the length of the transverse axis 
The length of the transverse axis is given by: \[ 2a = 2 \times 3 = 6 \]

Final Answer: The length of the transverse axis is 6.