Question

The length of the minor axis of the ellipse with foci \( (\pm 2, 0) \) and eccentricity \( \frac{1}{3} \) is:

• A. 2
• B. 3
• C. \( 2\sqrt{2} \)
• D. \( 4\sqrt{2} \)
• E. \( 8\sqrt{2} \)

Hint:

For an ellipse, use the formula \( c^2 = a^2 - b^2 \) and the given eccentricity to determine the minor axis.

Answer 1

Answer: (E)

For an ellipse, the relationship between the semi-major axis \( a \), semi-minor axis \( b \), and the focal distance \( c \) is given by: \[ c^2 = a^2 - b^2 \] The given foci are at \( (\pm 2, 0) \), so \( c = 2 \). The eccentricity \( e \) is defined as: \[ e = \frac{c}{a} \] Given that \( e = \frac{1}{3} \), we solve for \( a \): \[ \frac{2}{a} = \frac{1}{3} \Rightarrow a = 6 \] Using \( c^2 = a^2 - b^2 \): \[ 4 = 36 - b^2 \Rightarrow b^2 = 32 \Rightarrow b = 4\sqrt{2} \] The length of the minor axis is \( 2b = 8\sqrt{2} \).