Is there an equation to find the tangent of an inclined ellipse?

Question

There is a point (x,y) in the elliptical orbit inclined at theta angle. I know the x, y coordinates and the center coordinates and Major Minor. In this case, is there an equation to find the tangent line?

Answer 1

If $S$ is the center, $\theta$ the angle of inclination and $a,b$ the semi-axes lengths, the equation would be $$\frac{((x-x(S)) \cos{\theta}+(y-y(S)) \sin{\theta})^2}{a^2}+\frac{(-(x-x(S)) \sin{\theta}+(y-y(S))\cos{\theta})^2}{b^2}=1,$$ and the tangent at $(h,k)$ is given from the expanded form $Ax^2+2Bxy+Cy^2+2Dx+2Ey+F=0$ by the usual $Axh+B(xk+yh)+Cyk+D(x+h)+E(y+k)+F=0.$