The eccentricity of the curve represented by $ x = 3 (\cos t + \sin t) $, $ y = 4 (\cos t - \sin t) $ is:
The eccentricity \( e \) of a conic curve, parameterized by \( x(t) \) and \( y(t) \), is defined as: \[ e = \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \] Given the parametric equations: \[ x = 3 (\cos t + \sin t) \] \[ y = 4 (\cos t - \sin t) \] 1. Calculate the derivatives with respect to \( t \): \[ \frac{dx}{dt} = 3 (-\sin t + \cos t) \] \[ \frac{dy}{dt} = 4 (-\sin t - \cos t) \] 2. Determine \( \frac{dy}{dx} \) using the chain rule: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{4 (-\sin t - \cos t)}{3 (-\sin t + \cos t)} \] 3. Compute the eccentricity: \[ e = \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \] Upon simplification, the eccentricity is found to be \( \frac{\sqrt{7}}{4} \). Therefore, the correct answer is \( \frac{\sqrt7}{4} \).