# How to find the unit tangent, the principal normal and the curvature of $\frac{x^2}{9}+\frac{y^2}{4}=1$?

I need help with this problem:

Sketch the semi ellipse $$C$$ defined by $$\frac{x^2}{9}+\frac{y^2}{4^2}=1 \quad\quad\quad\quad x\leq0$$Assuming a "clockwise" orientation of $$C$$ (from $$(0,-2)$$ to $$(0,2)$$), find the unit tangent, the principal normal and the curvature to C at $$\left(\frac{-3}{\sqrt{2}},\frac{2}{\sqrt{2}}\right)\in C$$.

I parametrized the ellipse like this: $$x=3\cos t$$ and $$y=-2\sin t$$, where $$t\in[\frac{\pi}{2},\frac{3\pi}{2}]$$

After that I calculated the first and second derivative of the parametrization: $$f'(t)=(-3\sin t, -2\cos t)$$ $$f''(t)=(-3\cos t, 2\sin t)$$ Then I tried to calculate the unit tangent, the principal normal and the curvature at the given point.

For that, I found $$t$$ by doint $$\frac{-3}{\sqrt{2}}=3\cos t\Rightarrow t=\frac{3\pi}{4}$$ and $$\frac{2}{\sqrt{2}}=-2\sin t\Rightarrow t=-\frac{\pi}{4}$$ and since $$t\in[\frac{\pi}{2},\frac{3\pi}{2}]$$, I chose $$t=\frac{3\pi}{4}$$

(1) $$T(t)=\frac{f'(t)}{\Vert f'(t)\Vert}$$=$$\left(\frac{-3\sin t}{\sqrt{9\sin^2 t+4\cos^2 t}}, \frac{-2\cos t}{\sqrt{9\sin^2 t+4\cos^2 t}}\right)$$, thus $$T(\frac{3\pi}{4})=\left(\frac{-3\sqrt{2}}{\sqrt{26}},\frac{2\sqrt{2}}{\sqrt{26}} \right)$$.

(2) I tried to calculate the principal normal with the formula $$N(t)=\frac{T'(t)}{\Vert T'(t)\Vert}$$, but I really got stuck at calculating the norm of $$T'(t)$$. I found that $$T'(t)=\left(\frac{-3(2\cos t (9\sin^2 t+4\cos^2 t)-5\sin (2t)\sin t}{2(9\sin^2 t+4\cos^2 t)\sqrt{9\sin^2 t+4\cos^2 t}}, -\frac{-2\sin t(2(9\sin^2 t+4\cos^2 t)\sqrt{9\sin^2 t+4\cos^2 t})-5\sin (2t)\cos t}{(9\sin^2 t+4\cos^2 t)\sqrt{9\sin^2 t+4\cos^2 t}} \right)$$. I don't know if I'm correct. I tried to calculate the norm of that, but I couldn't.

(3) I need (2), so I can use $$\kappa(t)=\frac{\Vert T'(t)\Vert}{\Vert f'(t)\Vert}$$, or as one of the formulas that appear in my book $$\kappa(q)=\frac{f''(a)\cdot N(q)}{\Vert f'(a)\Vert^2}$$.

Please, I need help with calculating $$N(t)$$, also, I don't know if it is correct, but a friend of mine used the fact that $$N(t)\cdot T(t)=0$$ to calculate $$N(t)$$, how can I use this?