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

Hint:

Eliminate t by squaring and adding the scaled equations to reach the standard ellipse form.

Answer 1

The Correct Option is A

Step 1: Identify the curve
Squaring and adding the two scaled parametric equations removes t and gives an ellipse.

Step 2: Compare with the standard form
$\dfrac{x^2}{18} + \dfrac{y^2}{32} = 1$ has the larger denominator under $y^2$, so the major axis is vertical.

Step 3: Use the formula
$e = \sqrt{1 - \dfrac{18}{32}} = \sqrt{\dfrac{7}{16}} = \dfrac{\sqrt{7}}{4}$.

Final Answer:
The eccentricity is root 7 over 4. \[ \boxed{\text{(A) }\dfrac{\sqrt{7}}{4}} \]