The eccentricities of conic sections with one focus at the origin and the directrix corresponding and sketch a graph:displaystyle{e}={2}, directrix displaystyle{r}=-{2} sec{theta}.

How to find the position on ellipse (or hyperbola) arc if we know it's euclidean distance from given point and direction of movement?

Finding equation of a path in the plane ${\displaystyle y=z}$
What is the easiest way to see that the path ${\displaystyle \underset{―}{r}:\mathbb{R}\to {\mathbb{R}}^3:t\mapsto (\sin ,t,\cos ,t,\cos ,t)}$
traces out an ellipse in the plane ${\displaystyle y=z}$?
I think first rotating ${\displaystyle {\mathbb{R}}^3}$ by ${\displaystyle \frac{\pi }{4}}$ about the x-axis will help but I am not sure how to proceed.

Graph the lines and conic sections ${\displaystyle r=\frac{8}{4+\cos \theta }}$

Angle between normal vector of ellipse and the major-axis.
I am trying to derive the angle made between the major or x-axis and the normal vector of an ellipse of general shape ${\displaystyle x=a\cos \left(t\right),y=b\sin \left(t\right)}$ with the parameter t reffering to Ellipse in polar coordinates. I need to solve it for any angle t. From standard reasoning I find the normal vector by its definition and checked it with the page on mathworld from wolfram and works well. Then since I know 2 points, namely a point ON the shape and a point on the normal vector I derive the angle of interest to be ${\displaystyle \tan \left(\varphi \right)=\frac{a}{b}\tan \left(t\right)}$ Derivation. However this is very similar to the polar angle namely its simply the term a and b flipped. But when thinking about it I keep getting confused, am I correct or do I need the polar angle? If so where did I go wrong?
I also found Normal to Ellipse and Angle at Major Axis but this page confused me a bit, one idea I had was they use the polar angle vs the angle I am in need of ${\displaystyle \left(\varphi \right)}$ then I would indeed get by combing ${\displaystyle t={\tan }^{-1}\left(\frac{a}{b}\tan \left(\theta \right)\right)}$ and ${\displaystyle \varphi ={\tan }^{-1}\left(\frac{a}{b}\tan \left(t\right)\right)}$
${\displaystyle \varphi ={\tan }^{-1}\left(\frac{a}{b}\tan \left({\tan }^{-1}big\left(\frac{a}{b}\tan \theta big\right)\right)\right)={\tan }^{-1}\left(\frac{a^2}{b^2}\tan \theta \right)}$
My excuse for my rambling, I find these angles confusing...

How do you name the curve given by the conic ${\displaystyle r=\frac{4}{1+\cos \theta }}$?

The type of the conic section using the Discriminant Test and plot the curve using a computer algebra system. ${\displaystyle x^2-2xy+y^2+24x-8=0}$

Instructions:
Graph the conic section and make sure to label the coordinates in the graph. Give the standard form (SF) and the general form (GF) of the conic sections.
CIRCLES:
Center is at ${\displaystyle (2,-4).}$ the diameter's length is 6. The endpoints of the diameter is at ${\displaystyle (-1,-4)}$ and ${\displaystyle (3,6).}$