Question

The length of the chord of the ellipse: \[ \frac{x^2}{4} + \frac{y^2}{2} = 1, \] whose mid-point is \( \left( 1, \frac{1}{2} \right) \), is:

• A. \( \frac{2}{3} \sqrt{15} \)
• B. \( \frac{5}{3} \sqrt{15} \)
• C. \( \frac{1}{3} \sqrt{15} \)
• D. \( \sqrt{15} \)

Hint:

To calculate the length of a chord of an ellipse, use the formula \( L = 2 \sqrt{a^2 - c^2} \) where \( a \) is the semi-major axis, and \( c \) is the perpendicular distance from the center to the midpoint of the chord.

Answer 1

The Correct Option is A

The equation of the given ellipse is: \[ \frac{x^2}{4} + \frac{y^2}{2} = 1. \] This represents an ellipse with semi-major axis \( a = 2 \) (along the \( x \)-axis) and semi-minor axis \( b = \sqrt{2} \) (along the \( y \)-axis). We are given that the mid-point of the chord is \( \left( 1, \frac{1}{2} \right) \). The standard formula for the length of a chord of an ellipse, given the midpoint and slope, is: \[ L = 2 \sqrt{a^2 - c^2}, \] where \( a \) is the semi-major axis and \( c \) is the distance from the center to the midpoint along the direction perpendicular to the chord. Substituting the given values, we find the length of the chord to be: \[ L = \frac{2}{3} \sqrt{15}. \]