Question

The eccentricity of the curve represented by $ x = 3 (\cos t + \sin t) $, $ y = 4 (\cos t - \sin t) $ is:

• A. \( \frac{\sqrt{7}}{4} \)
• B. \( \frac{1}{16} \)
• C. \( \frac{\sqrt{7}}{3} \)
• D. \( \frac{\sqrt{8}}{4} \)

Hint:

For parametric curves, use the chain rule to find \( \frac{dy}{dx} \) and then calculate the eccentricity using the formula \( e = \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \).

Answer 1

The Correct Option is A

The eccentricity \( e \) of a conic curve represented parametrically by \( x(t) \) and \( y(t) \) is given by: \[ e = \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \] We are given: \[ x = 3 (\cos t + \sin t) \] \[ y = 4 (\cos t - \sin t) \] 
1. First, we find \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \): \[ \frac{dx}{dt} = 3 (-\sin t + \cos t) \] \[ \frac{dy}{dt} = 4 (-\sin t - \cos t) \] 
2. Then, we calculate \( \frac{dy}{dx} \) 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. Finally, we compute the eccentricity: \[ e = \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \] 
After simplifying the expression, we get the eccentricity as \( \frac{\sqrt{7}}{4} \). 
Thus, the correct answer is \( \frac{\sqrt7}{4} \).