For questions about parametric equations, their application, equivalence to other equation types and definition.

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3answers
49 views

express $\frac{\sin 3a}{\sin a}$ with only $\cos a$

How can I express $\frac{\sin 3a}{\sin a}$ while using only $\cos a$? Thanks in advance
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0answers
69 views

Solving a non-linear parametric equation

I am interested in solving a parametric equation where the unknown function is a function of time, and there is also an input. For example: $ y^{2}(t) + y(t) = \sin(t)$ I am coming from a signal ...
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2answers
89 views

How to prove parametric equation of a ellipse

The parametric equation of a ellipse is $$x=a \cos t\\y=b \sin t$$ It can be viewed as $x$ coordinate from circle with radius $a$, $y$ coordinate from circle with radius $b$. How to prove that it's ...
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0answers
30 views

Collinearity of three points on a curve.

In the realm of elliptic curves, the collinearity of three points is of a fundamental importance because this condition allows us to define on the curve a law of Abelian group, the study of which is ...
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2answers
34 views

How to solve this parametric linear equation?

How to solve this parametric linear equation? I need to find all numbers for $\alpha$ with which has a single, infinity or none solution. $$ \left[\begin{array}{rrr|r} \alpha & 1 & 0 ...
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2answers
51 views

How to use parametric equation/trigonometric identity to show an ellipse?

I have the equation $16x^2+25y^2=400$, and the parametric equation $(x,y)=(5\cos t, 4\sin t)$. If I plug in the parametric equation into the first equation, I end up with the trigonometric identity ...
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6answers
104 views

If $a^2 + b^2 = 1$, show there is $t$ such that $a = \frac{1 - t^2}{1 + t^2}$ and $b = \frac{2t}{1 + t^2}$

My question is how we can prove the following: If $a^2+b^2=1$, then there is $t$ such that $$a=\frac{1-t^2}{1+t^2} \quad \text{and} \quad b=\frac{2t}{1+t^2}$$
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2answers
56 views

Calculus problem of finding the equation of a line.

Find the equation of a line that passes through the origin, with positive slope, and its tangent to the parabola given by :$ y = x^2 - 2x + 2$ My approach to this problem was to differentiate the ...
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2answers
61 views

Parametric Trig Functions

A closed curve in the $(x, y)$-plane is represented by the functions $$x(θ)=\frac12(\cos \theta +\sqrt2 (\sin \theta))$$ $$y(θ)=\frac12(− \cos \theta +\sqrt2 (\sin \theta))$$ where the parameter ...
2
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2answers
51 views

How to find the points in which a given curve intersects itself?

Apologies in advance for my lack of knowledge with *tex. Hi everyone and thanks for any sort of help! I am given the following parametric curve: $(t^2\cos t, t^2\sin t,t^2), \text{where} -2\pi \le ...
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0answers
20 views

Parametrized curve with adjustable plateau

I am trying to create a parametrized curve. Basically I want a monotone curve through $(0, 0)$ and $(u, 1)$ with a plateau at $(ru, s)$ with $u\gt 0$ and $r,s\epsilon[0; 1]$, so my constraints are ...
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1answer
30 views

How to represent 2D lines (i.e. on $x$ and $y-axis$) on a 3D graph using either Cartesian ($z=f(x,y)$) or Parametric $(x,y,z)=f(u,v)$

How to represent 2D lines (i.e. on $x$ and $y-axis$) on a 3D graph using either Cartesian ($z=f(x,y)$) or Parametric $(x,y,z)=f(u,v)$ Hi, I've been working on a Simplex problem and would like to ...
2
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3answers
36 views

Find the derivative $dy/dx$ from the parametric equations for $x$ and $y$

Let \begin{cases} y=2t^2-t+1 \\ x=\sin(t) \end{cases} find $\frac{dy}{dx}$ Is this all that I need to do? $$\frac{4t-1}{\cos(t)}$$
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2answers
38 views

Parametric Equations. Find $\frac{dy}{dx}$ in terms of $x$

Find $\frac{dy}{dx}$ in terms of $x$ if the parametric equations of a curve are given by $x=e^{\sqrt{4t}}$ and $y=\sqrt{e^{6t}}$. My attempt, I found $\frac{dx}{dt}=\frac{e^{2\sqrt{t}}}{\sqrt{t}}$ ...
0
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1answer
21 views

Parametric Equations (Concavity)

The question is: A curve is defined by the parametric equations $$ x = t^2 + a $$ $$ y = t(t-a)^2 $$ Find the range of values for t in terms of a where the function is concave up? What I have ...
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0answers
42 views

I don't understand these directional vectors

I'm currently practicing for my final calculus exam in 20 days time, it has a vector section, here is my problem. When I need to find these parametric equations for either lines of planes, I need a ...
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2answers
54 views

Path of a cycloid

In this question, it's said that the path of a cycloid can be given as this parametric equation: $$\begin{align*}x &= r(t - \sin t)\\ y &= r(1 - \cos t)\end{align*}$$ and is shown here: ...
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1answer
87 views

Parameterization of a torus

Given that the parameterization of a torus is given by: $x(\theta,\phi) = (R + r\cos(\theta))\cos(\phi)$ $y(\theta,\phi) = (R + r\cos(\theta))\sin(\phi)$ $z(\theta,\phi) = r\sin(\theta)$ and the ...
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0answers
36 views

How do I detect collisions in a parametric equation$?$ (Lissajous curve)$?$

Let's say you have a simple parametric equation where $x= \sin(3t)$ and $y=\cos(7t).$ This is a pretty simple parametric equation that generates a relatively complicated Lissajous figure. You can ...
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3answers
67 views

Parametrization of a hypocycloid

How do I prove that a hypocycloid, which has equation $$x^{2/3} + y^{2/3} = a^{2/3}$$ can be parameterized by $$x = a\cos^3(\theta),\qquad y = a\sin^3(\theta)$$? The problem assumes that it is true, ...
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1answer
25 views

Parametric equation for $x²+4y² =1$, $x²+z^2=1$? ($y\ge 0, z\ge 0$)

In order to find this parametric equation, I would do: $$z=\sqrt{1-x²}\\x=\sqrt{1-4y²}\implies \\z = \sqrt{1-(1-4y²)} = \sqrt{1+4y²}$$ Then, if I choose $x=t$ as a parameter, I get: $$x=t, ...
3
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3answers
60 views

How to find parametric equation of the intersection $x²+y²+z²=2$ and $y=x$?

I know that if I substitute $y=x$ into $x²+y²+z²=2$ I get $$2x²+z²=2$$ which in some way gives me $$x²+\frac{z²}{2} = 1$$ which is an ellipse. My parametric equation goes from $(0,0,\sqrt{2})$ ...
0
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0answers
29 views

Parametrization of $x^2-y^2=1$; Why does the following only cover the “right-hand side”?

I have the parabola $x^2-y^2=1$. I am asked to first show that $\cosh(t)=x,\sinh(t)=y$ parameterizes the curve. This, I have done; substituting to the implicit function expression gives $1$. But I ...
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1answer
25 views

Find a Function That is in The Shape of the Given Boundary

John lives 2 miles north from a road, which is separated from John's house by a grove. If he walks from his house to the road along any straight line, the last mile of his walk is through the ...
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1answer
20 views

Different forms of representing curves

I am reading about mathematical representation of curves and have come across following points which I can't seem to understand : 1) Why Explicit representation cannot be used to represent closed ...
1
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1answer
39 views

Second degree parametric inequalities

I am asked to solve the parametric inequality and to find for which values of a every x is a solution. $$ (a+5)x^2 - 2x(a+1) + 2a - 4 \ge 0 $$ So in order every x to be a solution to the ...
0
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0answers
31 views

Cycloid angular parameter solution to an ODE for density fluctuations

I'm just reading over some Cosmology notes and there is a little ODE solve that I am not quite understanding. I have an equation of the form: $$ \ddot{R}=-\frac{GM}{R^{2}} $$ Integrating gives: $$ ...
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1answer
21 views

Length and revolved surface area of parametric curve

Can someone verify this for me?
2
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1answer
56 views

Creating a Parametric Equation

The problem is as follows: "In each of the following cases, you will be asked to write down a family of parametric curves that have the property that at $$t = 1$$ we have $$x'(t) = y'(t) = 0$$ but the ...
2
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1answer
17 views

Convergence of a sequence depending on parameter

My math teacher gave us this exercise for homework: Given $ a \in [-3, \infty), $ a given sequence $(x_n),$ with $ x_n = \left(\frac{a^n+2}{3^n+4}\right)$ Determine $a$ so that the sequence ...
2
votes
2answers
84 views

On the complete solution to $x^2+y^2=z^k$ for odd $k$?

While trying to answer this question, I was looking at a computer output of solutions to $x^2+y^2 = z^k$ for odd $k$ and noticed certain patterns. For example, for $k=5$ we have $x,y,z$, $$10, 55, ...
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2answers
48 views

Set the parametric equation of an arc with two points

Like the title says, I am looking for a method to find the parametric equation expressed in the form $\vec{r}=...\vec{i}+...\vec{j}$ of the arc that connects the points (2,0) and (1,2). I am asking ...
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1answer
19 views

Calculate the work of the force generated by an electric charge in movement

We know that the force generated by an electric charge that is located at the origin, on a charged particle at a point $(x,y,z)$ of position vector ...
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1answer
21 views

Moving of 5 units on a curve

If I am at a point (0, 0, 3) and a move of 5 units in the positive direction of $t$ on a curve definite by $$ \begin{align*} x &= 3\sin(t) \\ y &= 4t \\ z &= 3\cos(t)\\ \end{align*} $$ ...
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1answer
42 views

Domain and range of locus formed from parametric equations

\begin{cases} x = t^2 + 2t\\ y = 4(t+1)^2 \end{cases} Determine the cartesian equation of the locus? What is the domain of the locus? Note: I have found the cartesian equation: $y=4x+4$ I am just ...
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1answer
30 views

Parametrization of a cylinder that is parallel to x axis

The answer is no it does not matter. The surface is $y^2+z^2=4$, I parametrized it so: $\mathbf r=x \mathbf i +2\cos\theta \mathbf j + 2\sin\theta \mathbf k$ But Pauls Outline works through the ...
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1answer
72 views

find an appropriate parametrization for the given piecewise-smooth curve in $\mathbb{R^2}$

...for the curve $C$, which goes along the circle of radius $3$, from the point $(3,0)$ to the point $(-3,0)$, and then in a straight line along the $x$-axis back to $(3,0$). So I set the half-circle ...
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1answer
31 views

Eliminate the parameter to find a Cartesian equation of the curve.

I got $y=x^\frac{7}{2}$ as the cartesian equation for the following parametric equations, but it is showing up as incorrect. Can anyone explain to me what I did wrong? Thank you!
2
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1answer
29 views

Compute the length of a parametric curve.

It seems like I am not using the good process to compute the length of a given parametric curve. I am not sure if it's inside my calculations or if the steps I use are not correct. The equation of ...
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0answers
8 views

Find a parametric representation for an implicit function f(x,y,z)=0

I encountered a very interesting implicit function: $ z \exp \left[ (x-0.5-e^{z-y})^2+y^2-0.2z+3 \right] = \sin \left[ (xz-0.5)^2+2xy^2-0.1z \right]$ I wonder if there is any ways to find a ...
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1answer
31 views

Length of the curve $ \;\;x=3\cos\!\left(6t\right), \;\;y=18t+3\sin\!\left(6t\right), \, \;\; \;\; 0 \le t \le \frac{\pi }{6}\;\;$

The length $\;L\;$ of the curve C given by $\displaystyle \;\;x=3\cos\!\left(6t\right), \;\;y=18t+3\sin\!\left(6t\right), \, \;\; \displaystyle \;\; 0 \le t \le \frac{\pi }{6}\;\;$ is found by ...
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2answers
61 views

How to handle this curve? [closed]

I started with differentiation of all three coordinates of this parametrically given curve. I want to show that the respective curve has related equation of the plane and also to prove that it a ...
0
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1answer
34 views

Tangents to Parametrized Curves

I need to find the equation for the line tangent to the curve: $$ x = 2\cos t, y = 2\sin t, t = \frac \pi4 $$ I also need to find the value of: $$ \frac{d^2y}{dx^2} $$ I'm not too sure what to do.. ...
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2answers
45 views

Find Cartesian equation from Parametric Equations Including Sec and Tan

Need to find the cartesian equation from: $$ x = sec^2t - 1 , y = tan t, -\frac\pi2 \lt t \lt \frac \pi2 $$ With sin and cosine I use the unit circle, but I don't know what to do with sec and ...
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1answer
34 views

Finding Cartesian Equation giving Parametric Equations

The equations are: $$ x = cos(\pi - t), y = sin(\pi - t), 0 \le t \le \pi$$ I don't really understand what to do. On the last problem I had: $$ x = cos2t, y = sin2t, 0 \le t \le \pi $$ and I just ...
1
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3answers
243 views

Show that the curve has two tangents

I'm a little stuck on a math problem that reads as follows: Show that the curve $x = 5\cos(t), y = 3\sin(t)\, \cos(t)$ has two tangents at $(0, 0)$ and find their equations What I've Tried $ ...
1
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0answers
22 views

Is there a program that receives as input your drawing of a curve and outputs a parametric curve tracing it (reasonably close)?

From what I know, B-Splines is the closest thing that we have to drawing curves and having them defined by the computer. I have some B-Spline code that does this interactively. However, those are a ...
0
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0answers
38 views

Fourier Series and epicycles - How to extract the radii and angular velocities from the Fourier Series expansion of a function.

NOTE: I am attaching Mathematica code for those who may want to check it out and understand what I'm asking for. The rest of the question is pretty mathematical in nature, I'll also try the ...
2
votes
1answer
20 views

Finding c in parametric quadratic equation

I tried searching before posting this, but couldn't find anything. I have this parametric* equation : $ x^2 + 54x + 5a^2 = 0 $ They are asking me to find the values of a for which the roots of the ...
0
votes
2answers
25 views

Express the region $D=\{(x,y): x^2\leq y \leq x^2+x^3, x>0 \}$ as the union of cubic curves

Let $D=\{(x,y): x^2\leq y \leq x^2+x^3, x>0 \}$ I know the family of curves $\gamma(t)=x^2+tx^3$ belong to D, for $t\in [0,1]$. It is true that for every $(x,y)\in D$ there exist a unique ...