Question

The foci of a hyperbola are (8,3) and (0,3) and eccentricity is $4/3$. Then the length of the transverse axis is:

• A. $32/3$
• B. 4
• C. 8
• D. $8/3$
• E. 6

Hint:

Always identify if the transverse axis is horizontal or vertical. Here, $y=3$ is constant, so it is horizontal.

Answer 1

Answer: (E)

To find the length of the transverse axis of the hyperbola, we are given:

• The coordinates of the foci: (8, 3) and (0, 3)
• The eccentricity, \(e = \frac{4}{3}\)

The standard equation for a horizontal hyperbola centered at \((h, k)\) is:

\(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\)

For such a hyperbola, the distance between its center and each focus is denoted as \(c\), and \(c^2 = a^2 + b^2\).

The eccentricity is defined as \(e = \frac{c}{a}\).

First, calculate the distance between the foci (8, 3) and (0, 3):

\(c = \frac{8 - 0}{2} = 4\)

Here, the center of the hyperbola is at the midpoint of the foci:

\(h = \frac{8 + 0}{2} = 4\) and \(k = 3\)

We have \(e = \frac{c}{a}\), thus \(\frac{4}{3} = \frac{4}{a}\).

Solving for \(a\) gives you:

\(a = 3\)

The length of the transverse axis is \(2a\):

\(2 \cdot 3 = 6\)

Thus, the correct answer is 6.