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 equation of the ellipse is given by: \[ \frac{x^2}{2} + \frac{y^2}{4} = 1. \] This is in the standard form of an ellipse with semi-major axis \( a = 2 \) and semi-minor axis \( b = \sqrt{2} \). The formula for the length of a chord with midpoint \( (x_0, y_0) \) in an ellipse is: \[ L = 2 \sqrt{b^2 - \left( \frac{b^2 x_0^2}{a^2} \right)}. \] Substitute the given values \( a = 2 \), \( b = \sqrt{2} \), and \( x_0 = \frac{3}{2} \) into the formula: \[ L = 2 \sqrt{2 - \left( \frac{2 \times \left(\frac{3}{2}\right)^2}{2} \right)}. \] Simplify the equation: \[ L = 2 \sqrt{2 - \left( \frac{2 \times \frac{9}{4}}{2} \right)} = 2 \sqrt{2 - \frac{9}{4}} = 2 \sqrt{\frac{8}{4} - \frac{9}{4}} = 2 \sqrt{\frac{-1}{4}}. \] Since the expression inside the square root is negative, this implies that the point \( ( \frac{3}{2}, 0) \) does not lie on the ellipse, and thus no real chord exists with this midpoint. Thus, there is no valid solution for the length of the chord in this case.