The eccentricities of conic sections with one focus at the origin and the directrix corresponding and sketch a graph:

Alyce Wilkinson
2021-03-09
Answered

The eccentricities of conic sections with one focus at the origin and the directrix corresponding and sketch a graph:

You can still ask an expert for help

Neelam Wainwright

Answered 2021-03-10
Author has **102** answers

Since

Now consider the directrix,

Now comparing whith

Therefore equation is,

Now the graph is,

asked 2022-04-21

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

asked 2022-04-06

Finding equation of a path in the plane $y=z$

What is the easiest way to see that the path$\underset{\u2015}{r}:\mathbb{R}\to {\mathbb{R}}^{3}:t\mapsto (\mathrm{sin},t,\mathrm{cos},t,\mathrm{cos},t)$

traces out an ellipse in the plane$y=z$ ?

I think first rotating$\mathbb{R}}^{3$ by $\frac{\pi}{4}$ about the x-axis will help but I am not sure how to proceed.

What is the easiest way to see that the path

traces out an ellipse in the plane

I think first rotating

asked 2021-08-12

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

asked 2022-03-15

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$x=a\mathrm{cos}\left(t\right),y=b\mathrm{sin}\left(t\right)$ 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 $\mathrm{tan}\left(\varphi \right)=\frac{a}{b}\mathrm{tan}\left(t\right)$ 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$\left(\varphi \right)$ then I would indeed get by combing $t={\mathrm{tan}}^{-1}\left(\frac{a}{b}\mathrm{tan}\left(\theta \right)\right)$ and $\varphi ={\mathrm{tan}}^{-1}\left(\frac{a}{b}\mathrm{tan}\left(t\right)\right)$

$\varphi ={\mathrm{tan}}^{-1}\left(\frac{a}{b}\mathrm{tan}\left({\mathrm{tan}}^{-1}big\left(\frac{a}{b}\mathrm{tan}\theta big\right)\right)\right)={\mathrm{tan}}^{-1}\left(\frac{{a}^{2}}{{b}^{2}}\mathrm{tan}\theta \right)$

My excuse for my rambling, I find these angles confusing...

I am trying to derive the angle made between the major or x-axis and the normal vector of an ellipse of general shape

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

My excuse for my rambling, I find these angles confusing...

asked 2022-02-10

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

asked 2021-01-05

The type of the conic section using the Discriminant Test and plot the curve using a computer algebra system.

asked 2021-08-10

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$(2,\text{}-4).$ the diameter's length is 6. The endpoints of the diameter is at $(-1,\text{}-4)$ and $(3,\text{}6).$

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