1. ## Eccentric angle

I'm stuck on this question:

The eccentric angle corresponding to the point (2,1) on the ellipse with equation x^2+9y^2=13 is theta (don't know how to do the symbol). Find tan theta.

When I attempted the question I got 1/2, but the answer is 3/2. Please can someone explain why.

2. Originally Posted by free_to_fly
I'm stuck on this question:

The eccentric angle corresponding to the point (2,1) on the ellipse with equation x^2+9y^2=13 is theta (don't know how to do the symbol). Find tan theta.

When I attempted the question I got 1/2, but the answer is 3/2. Please can someone explain why.
Hello,

parametrize the equation of an ellipse with it's center in the origin, the major semiaxis a and the minor semiaxis b:
$\displaystyle \left|\begin{array}{l}x=a \cdot \cos(\theta) \\ y = b \cdot \sin(\theta)\end{array}\right.~\implies~\left|\beg in{array}{l}\frac xa= \cos(\theta) \\ \frac yb = \sin(\theta)\end{array}\right.$

Now divide the second equation by the first equation:

$\displaystyle \frac{\sin(\theta)}{\cos(\theta)}=\tan(\theta)=\df rac{\frac yb}{\frac xa}=\frac{a \cdot y}{b \cdot x}$ Plug in the values you already know:

$\displaystyle \tan(\theta)=\frac{\sqrt{13} \cdot 1}{\frac13 \cdot \sqrt{13}\cdot 2} = \frac32$