Question

Find the length of the chord whose midpoint is \( \left( \frac{3}{2}, 0 \right) \) of the ellipse \[ \frac{x^2}{2} + \frac{y^2}{4} = 1. \]

Answer 1

The ellipse's equation is \[\frac{x^2}{2} + \frac{y^2}{4} = 1.\]This standard form indicates a semi-major axis \( a = 2 \) and a semi-minor axis \( b = \sqrt{2} \).The formula for the length of a chord with midpoint \( (x_0, y_0) \) on an ellipse is \( L = 2 \sqrt{b^2 - \left( \frac{b^2 x_0^2}{a^2} \right)} \).Substituting \( a = 2 \), \( b = \sqrt{2} \), and \( x_0 = \frac{3}{2} \) yields:\[L = 2 \sqrt{2 - \left( \frac{2 \times \left(\frac{3}{2}\right)^2}{2} \right)}.\]Simplification leads to:\[L = 2 \sqrt{2 - \left( \frac{2 \times \frac{9}{4}}{2} \right)} = 2 \sqrt{2 - \frac{9}{4}} = 2 \sqrt{\frac{8}{4} - \frac{9}{4}} = 2 \sqrt{\frac{-1}{4}}.\]The negative value under the square root signifies that the point \( ( \frac{3}{2}, 0) \) is not on the ellipse, meaning no real chord exists with this midpoint.Consequently, there is no valid solution for the chord length in this scenario.