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

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2answers
716 views

Reverse direction of parametric equation

For the graph $y = \sqrt{x}$ the normal parametric equations would $x = t^2$ and $y = |t|$. However, the direction for that graph would be going from infinity to zero when $t \leq 0$ and zero to ...
2
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0answers
23 views

Solving a Diophantine equation in three variables as a parametric equation in one variable

Let’s say that $a$, $b$, and $c$ are integers such that $$(b^2+2)^2=(a^2+2c^2)(bc-a). \tag{$\star$}$$ By brute force search, I think I’ve discovered that $$(a,b,c)=(5d+1,3d+1,d+2), \qquad ...
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2answers
22 views

parametric polar equation of a circle

I discovered that Mac's Grapher has a parametric polar mode, i.e. where $r$ and $\theta$ can be specified in terms of a parameter, usually $t$. I am attempting to convert the generic equation for a ...
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0answers
38 views

Intersection of a smooth plane curve and a circle

Let $\gamma(t)=(x(t),y(t)):[0,2\pi] \rightarrow \mathbb{C}$ be a simple and closed $C^1$-curve. Prove that there is a small circle that intersects $\gamma$ only at two points?
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1answer
23 views

Why are the parameterizations of the circle in cartesian coordinates defined in open intervals?

The parameterization of the circle in rectangular coordinates is given by the the following functions $$ y = g_1(x) = \sqrt{(1-x^2)} \\ y = g_2(x) = -\sqrt{(1-x^2)} \\ x = g_3(y) = \sqrt{(1-y^2)} \\ ...
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4answers
295 views

How did the answer key get $h=40-2r$?

A cone has radius of $20\ \rm cm$ and a height of $40\ \rm cm$. A cylinder fits inside the cone, as shown below. What must the radius of the cylinder be to give the cylinder the ...
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0answers
28 views

Parametric Equation of Elliptical Cycloidal Sine Curve

I am trying to find the parametric equations of a cycloidal curve, which, instead of using a circle, uses an ellipse to oscillate around a base circle. Below are equations of the standard, circular ...
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3answers
89 views

Parametrization of the intersection of two given surfaces

Find a parametrization of the intersection between the two curves $z=x^2-y^2$ and $z=x^2+xy-1$. I figure I should set them equal to each other but I'm not sure where to go from there: $$x^2-y^2 = ...
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2answers
22 views

Maximum Height is giving me negative

Hey guys for this parametric equations its giving me negative Question is: A dart is thrown from a point 5 feet above the ground with an inital velocity of 58 ft/sec and angle of elvation of 41∘. ...
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1answer
24 views

Parametric Problem: Throwing a Dart <Test Review>

Yup it's me ... Parametrics, who would have thought xD! Anyways, again ... I am doing review and I really need this grade to get an A in math class; that's why I am asking questions here. And you guys ...
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2answers
42 views

Parametric Problem: Kicking a football (Getting Ready For Test)

As I told you on title I'm getting ready for a test. I have this Test-Review Problem.... A football is kicked from the ground with an initial velocity of $28$ ft/sec at an angle of $28^\circ$. How ...
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1answer
26 views

How to fit a set of 3D points to a helical curve?

suppose I have a set of points in $\mathbb{R}^3$, and I want to find an arbitrary helix which best approximates these points. An arbitrary helix in $\mathbb{R}^3$ can be parametrized as ...
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1answer
504 views

Gradient function of a circle

The parametric equations of a circle $C$ are: \begin{align*} x&=2+\dfrac{13}{5\sqrt{2}}\cos t\\ y&=1+\dfrac{13}{5\sqrt{2}}\sin t \end{align*} for $t\in[0,2\pi]$. I am stuck on this part: Find ...
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1answer
35 views

Parametric Problem

i have a question on parametric.. The question states A vector equation $(x,y) = (2,-1) + t(3,2)$. Write as a parametric equation. Show a table with x,y values. Sketch a picture of vector ...
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3answers
393 views

Line integral around intersection of sphere and plane

The unit sphere is intersected by the plane x + y = 1. Find the line integral of F = around the intersection. $\int\int\nabla$x$F\cdot$ n dA the unit normal vector is easily found by looking at the ...
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1answer
28 views

Parametric Equations Problem

Im back! Um, i have a simple question im trying to get ready for test after 5 days.. I slacked of sadly :( on math, so i have to pick up my skills.. On my test review i have this question: The ...
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0answers
7 views

How can I find the necessary speed and speed of rotation for a problem from a parametric equation?

I have been given the following questions for a project that I am currently working on: Questions 1 to 8 I have completed questions 1 through 6 but have no idea how to do questions 6 or 7 after ...
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0answers
16 views

Heuristics for putting $f(x_0,x_1,\ldots,x_n)=0$ into parametric form?

Suppose I have an implicit equation: \begin{equation} f(x_0,x_1,\ldots,x_n)=0 \end{equation} Which might be 'paramaterizable'; i.e. put it into the form: \begin{align} x_0 &= ...
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1answer
633 views

Parametrization of the intersection of a cone and plane.

EDITED with new progress updates. As the title states, I'm trying to parametrize the intersection of a cone and a plane. The equations are: $z^2 = 2x^2+2y^2$ and $2x+y+3z=4\implies ...
2
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0answers
76 views

Pursuit Curve, Parametric Equation

So its a classic problem: Object $A$ starts at the origin $(0,0)$ and moves straight up the $y$ axis with a speed $v$. Object $B$ starts at point $(1,0)$, always moves towards object $A$ and has a ...
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2answers
43 views

Conic Sections: Hyperbola (Finding the Locus)

This is a multipart question so bear with me until I get to the part where I am stuck on. $H$: $xy=c^2$ is a hyperbola (i) Show that $H$ can be represented by the parametric ...
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0answers
28 views

Rotate an Ellipse

$x = h + a \cos(φ) \cos(θ) + b \sin(φ) \sin(θ)$ $y = k + b \cos(φ) \sin(θ) - a \sin(φ) \cos(θ)$ Hi, I have basic question of parametric equation for ellipse. I'm trying to rotate horizontal ellipse ...
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1answer
42 views

Find the point where the curve $\big(x(t),y(t)\big)=(t^2-1,t^3-t)$ crosses itself

Consider the curve defined by $x(t)=t^2-1$ and $y(t)=t^3-t.$ Find the point where the curve crosses itself. I know that the curve will cross itself if there are two distinct values, say $t_1$ and ...
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1answer
22 views

Finding algebraic curve satisfying given parameterization

Is there an easy way to find an algebraic curve that satisfies a given parameterization? Specifically, I am talking about the following parameterization: $$ x=z(1-z),\hspace{10pt} y=\sum_{n=1}^r ...
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0answers
42 views

Given any parametric curve, finding its general form?

I'll illustrate the problem I'm trying to solve with an example. Let's consider the equations $$ x = \cos (t) $$ $$ y = \sin (t) $$ We know that these are a parametric form of the unit circle. In ...
2
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4answers
106 views

How does one parameterize $x^2 + xy + y^2 = \frac{1}{2}$?

Parameterize the curve $C$ that intersects the surface $x^2+y^2+z^2=1$ and the plane $x+y+z=0$. I have this replacing equations: $$ x^2+y^2+(-x-y)^2=1$$ and clearing have the following: $$ ...
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1answer
66 views

Integral of logarithmic fuctions with parameter

Hello I am solving an integral with a natural logarithm that has a parameter. Let say $I(a)=\int_0^\pi\ln(1-2a\cos(x)+a^2)dx$ Then for differentiation under integral sign and that yields $I'(a)= ...
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0answers
20 views

How to find the limits of integration for parametric

In this question: Find the area bounded by: $x=\ln(t)$, $y=\frac{t-3}{t-1}$, $3\leq x \leq 5$, and by the $x$-axis (it is above the $x$ axis). I solved the integration parametric curve, $3\ln(t) ...
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2answers
2k views

Parametrization for the curve on cylinder $y = 7 - x^4$ that passes through the point $(0, 7, -3) $when t = 0 and is parallel to the xy-plane

Can you help me? So far I have turned $y = 7-x^4$ into $\langle1, 1, 0\rangle$ and used it to make the equation $L = (0, 7, -3) + t(1, 1, 0)$. I know this is wrong, but I just don't know what, and I ...
0
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1answer
36 views

How do parametric equations work?

I was given a graph like this in my exam. Its defined para-metrically by x=c^2 and y =c^3. It won't help me now but could someone explain this to me why I have two seemingly different lines I know ...
3
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1answer
25 views

Finding minimum plus maximum of $g(a)=\int_{0}^{\pi/2} |\sin 2x-a\cos x|dx$

Let $$g(a)=\int_{0}^{\pi/2} |\sin 2x-a\cos x|dx,\quad a\in[0,1].$$ If $L$ and $M$ are the minimum and maximum values of $g(a)$ for all $a\in [0,1]$. Find the value of $L+M$. The first thing ...
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1answer
26 views

Paramteric Curves and the exponents of $\cos$/$\sin$/$\tan$

Lets say we have the curve $\frac x7=\cos^7t$, $\frac y7=\sin^7t$ Now I know that $\sin^2x+\cos^2x=1$. So $\cos^2=(\frac x7)^{\text{some exponent}}$. What is that exponent? How do you work it out?
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1answer
38 views

Parametric curve, Write it in cartesian form, giving $y$ explicitly in terms of $x$ [closed]

$x=4\ln(t)$ $y=2t$ Write it in cartesian form, giving $y$ explicitly in terms of $x$.
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1answer
20 views

Parametric Equation of Cycloidal Sine Curve

I am trying to find the parametric equation of a sine curve, which oscillates around a circle as it's $x$-axis. I have done preliminary approximations using Epicycloid parametric equations for the top ...
0
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2answers
33 views

An equation with a parameter

Given the equation $(|x+1|+|x-a|)^2-2(|x+1|+|x-a|)+4a-4a^2=0$ find all possible $a$ such that this equation has only one solution. I wanted to solve it like this: ...
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1answer
27 views

2D parametric equation for an arc between two points with a start angle

What's a parametric equation (eg. $(x,y)=(\cos(t \cdot 2\pi),\sin(t \cdot 2\pi)$ plots a circle where $t$ is the 'time' along the circle) that draws an arc between the two points $(x_0,y_0)$ and ...
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1answer
267 views

Parametric integration negative area?

I know there is a question very similar to mine already here Why does using an integral to calculate an area sometimes return a negative value when using a parametric equation? , but I am still a bit ...
11
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4answers
380 views

Horizontal tangent line of a parametric curve

Suppose $x=t^2,y=t^3$ is a parametric curve. Here's a quote from my textbook: The origin, which corresponds to $t=0$, is a singular point of the parametric curve, because $dx/dt=2t,dy/dt=3t^2$ are ...
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1answer
21 views

How to find parametric equation between two points in line integral?

[In this example how can we find parametric equations of x and y.] [1] [question]: http://i.stack.imgur.com/lTOnW.png [1] [Solution]: http://i.stack.imgur.com/l8ao7.jpg
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1answer
18 views

Convert line parametrization into two equations

Consider the following parametrization on $\mathbb{R}^3$ $$g(t) = (t^2,t\cos(t),t\sin(t))$$ This is a line, and as such can be characterized by two equations. I already found the first one to be ...
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0answers
13 views

Derive the parametric form of the locus of point where difference between distance to two points is constant

Given two points $P_1=(x_1,y_1)$ and $P_2 = (x_2,y_2)$, the locus of the point whose (signed) difference between the distance to the two points is a constant $\Delta$ is one branch of a hyperbola ...
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2answers
34 views

Parametric version of a simple equation

I have a simple relation that I need to plot in a plane. I could do it, but I believe that I don't get the best way. A plane curve is defined implicitely by the following equation : ...
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1answer
23 views

Parametric Equations: Having trouble with finding two tangents (Calculus) [duplicate]

curve C defined by these parametric equations $x = t^3 - 3t^2$ $y = t^3 - 3t$ I need to find the equations of two tangents at the point $(-4,2)$ I have attempted to solve this problem myself but I ...
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1answer
20 views

Deriving parametric equations for a cubic equation

I've been looking at elementary cubic equations for curves and seem to understand them well enough. Going the other way and driving parametric equations has been mystifying. For example: given a ...
2
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0answers
26 views

Graphing/visualizing a complex parametric plot without using mathematica

I am trying to visualize the parametric plot in $\mathbb{C}$ of the curve $\gamma$ defined for $t\in[-\infty,\infty]$ as $$\gamma(t)=\exp\left(-t^{2}+\frac{t}{\sqrt{1+t^2}}i\right).$$ I think I find ...
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1answer
37 views

What's the name of this simple, closed, planar curve?

A simple, closed, planar curve can be given by the following parametric function: $$ \gamma(t)=\left(\cos t,\sin t+\frac{\sin^2t}{2}\right) $$ This function on $t=0$ to $t=2\pi$ gives the following ...
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0answers
26 views

How to determine the smallest period of a parametric curve?

Consider the polar function $r(\theta) = \sin(3\theta)$, and the parameterization of its graph given by $x = \sin(3\theta)\cos(\theta), \;y=\sin(3\theta)\sin(\theta)$. Upon inspection, one can observe ...
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votes
2answers
4k views

Finding a vector parametric equation given P and Q equations?

Find a vector parametric equation $r⃗(t)$ for the line through the points $P=(3,5,4)$ and $Q=(1,4,7)$ for each of the given conditions on the parameter $t$. If $r⃗ (0)=(3,5,4)$ and $r⃗ (7)=(1,4,7)$, ...
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1answer
28 views

Finding parametric equations of rectangular equation

Is there a general process to follow when finding the parametric equations of a normal rectangular equation ? I know that one rectangular equation might have many parametric equations, but are there ...
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0answers
71 views

Polynomial parametrization of a quadric with two given points

Let $X^1, X^2 \in \mathbb{R}^3$ be two distinct points of the quadric surface defined by the implicit function $$ \phi(X)= X^T\cdot A\cdot X + b^T \cdot X+c=0, $$ where and $A$, $b$ and $c$ are ...