Question

If \( (4, 3) \) and \( (12, 5) \) are the two foci of an ellipse passing through the origin, then the eccentricity of the ellipse is:

• A. \( \frac{\sqrt{13}}{9} \)
• B. \( \frac{\sqrt{13}}{18} \)
• C. \( \frac{\sqrt{17}}{18} \)
• D. \( \frac{\sqrt{17}}{9} \)

Hint:

The eccentricity of an ellipse is the ratio of the distance from the center to the foci to the length of the semi-major axis.

Answer 1

The Correct Option is D

Step 1: Find the distance between the foci. The distance between the foci \( (4, 3) \) and \( (12, 5) \) is given by: \[ d = \sqrt{(12 - 4)^2 + (5 - 3)^2} = \sqrt{8^2 + 2^2} = \sqrt{64 + 4} = \sqrt{68}. \] Step 2: Use the formula for eccentricity. For an ellipse, the eccentricity \( e \) is given by: \[ e = \frac{c}{a}, \] where \( c \) is the distance from the center to the foci, and \( a \) is the length of the semi-major axis. Since the ellipse passes through the origin, we know that \( c = \frac{d}{2} = \frac{\sqrt{68}}{2} = \sqrt{17} \). Thus, the eccentricity is: \[ e = \frac{\sqrt{17}}{9}. \] Thus, the correct answer is: \[ \boxed{4}. \]