Calculating the eccentricity for the ellipse.

Question

I am trying to calculate the eccentricity of an ellipse given by the equations: \begin{align*} \begin{cases} x(t) = 2\cos(t) + 1,\\\\ y(t) = 3\sin(t)- 2 \end{cases} \end{align*} where $t\in[0,2\pi]$.

Any help would be appreciated.

Thank you.

Answer 1

HINT

According to the fundamental trigonometric identity $\cos^{2}(\theta) + \sin^{2}(\theta) = 1$, the following relation holds: \begin{align*} \frac{(x-1)^{2}}{4} + \frac{(y+2)^{2}}{9} = 1 \end{align*}

Can you take it from here?

Answer 2

Hint: The points on that kind of ellipse can be expressed as $$ \left(\frac{x - x_0}{a}\right)^2 + \left(\frac{y - y_0}{b}\right)^2 = 1 $$