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

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2answers
67 views

Convert between parameteric ellipse equations

I have the parametric equation of an ellipse in this form: $$x(t)= a\cos(t)$$ $$y(t)=b\cos(t+\phi)$$ It's an ellipse centred about the origin, with a tilt angle. So three parameters. How can I ...
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0answers
40 views

Intersection between a parametric equation and a linear equation

2Consider the parametric functions $f_1, f_2$ with $$f_1(x) = 3(60-x)\cdot \sin(3x)$$ and $$ f_2(x) = 3(60-x)\cdot \cos(3x).$$ Suppose you have a linear function: $$f_3 (x) = 1.5 x$$ How does one ...
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2answers
136 views

How can I know whether the airplanes collide by using parametric equations

Recall that a line hes equation y=mx+c. Suppose one airplane moves along the line y=2x+3 while the other airplane moves along the line y=3x-2. By plotting a graph, even though the lines are intersect, ...
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1answer
462 views

Converting between explicit function and parametric function

Given an explicit function $y = f(x)$, how to convert it to the respective parametric functions $x = f_1(t)\; y = f_2(t)$? Given parametric functions $x = f_1(t)\; y = f_2(t)$, how to obtain the ...
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1answer
23 views

How to parametrise a parabola with a specific domain

What would be the best method to find the parametric equations for the parabola $y = (x-2)^2$ over a given domain of $(2 ≤ t ≤ 5)$? The figure I've been given has the parabola starting from $(2,0)$ ...
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1answer
52 views

How many solutions depending on the parameter (augmented matrix?)

I have to find how many solutions have got the following equations, depending on p parameter? $ \begin{bmatrix} 5 & p & 5 \\ 1 & 1 & 1 \\ p & p & 2 \end{bmatrix} $ $ \...
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2answers
58 views

Differentiate the parametric function and find $dy/dx$ and $d^2y/dx^2$

Differentiate the parametric function and find $\frac{\mathrm dy}{\mathrm dx}$ and $\frac{\mathrm d^2y}{\mathrm dx^2}$ in terms of "$t$" when: $ x = \frac{1}{t-1}$ and $y = \frac{1}{t+1}$ I have ...
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2answers
88 views

Showing that the image of a curve lies on a surface?

I am looking for an intuitive explanation to a problem in one of my practice tests. I'm given a parameterized curve from $\Bbb R$ to $\Bbb R^3$, called ${\bf r}(t) = (\sin t \cos t, \cos^2 t, \cos t)$....
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2answers
92 views

Investigating the “Wigglyness” of a 2D-Parametric Curve

I am looking to quantify the (for a lack of a better term) "wigglyness" of a parametric curve. The particular set of curves that I am looking at come from cubic-spline interpolation on a set of points ...
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2answers
46 views

Parametric equations: Finding the ordinary equation in $x$ and $y$ by eliminating the parameter from parametric equations

I am having difficult time solving the following equation: Eliminate the parameter from the parametric equations: $x=\frac{3t}{1+t3}$ and $y = \frac{3t^2}{1+t3}$ where $t \ne−1$ and hence find an ...
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1answer
44 views

Parametrizing motion of string around rod

I'm trying to solve this problem with calculus. A string is wound symmetrically around a circular rod. The string goes exactly 4 times around the rod. The circumference of the rod is 4 cm. and its ...
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0answers
101 views

parsimonious definition of a zig-zag function between two bounds

Suppose I have some strictly increasing function $f:[0,b]\to[0,b]$ with $0<b<1$, $f(x)<x$ and $f'(b)=\frac{1-f(b)}{1-b}$ (i.e. tangent to the secant line to $(1,1)$). Now imagine a 'tunnel' ...
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1answer
36 views

Line integral of a vector field? [closed]

First of all, sorry for the sketch.. I would be glad if you show me another ways to improve this drawn. I'm studying Line integrals, and this question is really boring me. Please, can someone put ...
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1answer
34 views

Parabola Question - simultaneous equations?

I'm having trouble with the second part of this question. I can do the first part by finding the normal at P and where it intercepts with U and then for the second part i've substituted each point ...
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1answer
87 views

Locus of solution of an ODE system

I have an ODE system $$ \ddot p = \frac{ p \left( {2p - 4} \right) }{{p - 4}}{{\dot q }^2 } \\ \ddot q = \frac{{3p - 8}}{{p - 4}}\dot q \dot p $$ Short of finding closed-form expressions for $p\left(...
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2answers
29 views

Use the discriminant to show that $mx−y + m^2 = 0$ touches the parabola $x^2 =−4y$, for all values of m.

Use the discriminant to show that $mx−y + m^2 = 0$ touches the parabola $x^2 =−4y$, for all values of m. I attempted to solve by letting them both equal each other, but it didn't work. How do I do ...
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2answers
39 views

Find, in terms of $s$,the coordinates of the point where this normal cuts the curve again.

a) Find the equation of the normal at the point $(2s,\frac{2}{s})$ to the curve whose parametric equations are $x=2s,y=\frac{2}{s}$ b) Find, in terms of $s$,the coordinates of the point where ...
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2answers
309 views

Gradient vector of parametric curve

I have ellipse $$(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$$ Gradient is $$(\frac{2x}{a^2}, \frac{2y}{b^2})$$ How I can obtain this vector from parametrization of my curve? Let I know only $$(x, y) = (a ...
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2answers
288 views

Radius of a wheel based on parametric equations

I am working on a question and I don't have the slightest idea where to begin. Any nudge in the right direction would be very helpful. Here is the question: A bicycle wheel has radius R. Let P be ...
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4answers
70 views

Eliminate $t$ to give an equation that relates $x$ and $y$

I am having problems understanding how to solve the following parametric equation. I have achieved an answer, but am unsure if my answer is correct or not. Eliminate t to give an equation that ...
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1answer
27 views

If PQis a focal chord, show that the interval RU is parallel to the axis of the parabola.

For part (c) of question thirteen am I only required to find the gradient of RU and prove that is it zero? This is how I have interpreted this question. ANY help on the matter is much appreciated ...
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1answer
45 views

Parametric Curve Tangent Equations

Let a curve be given in the parametrized form by: $r(t) = (2\cos t, 2\sin t), 0 \leq t \leq 2\pi$ Find the equations of the tangents to the curve at each of its points $(X_0, Y_0)$. Having gone ...
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2answers
36 views

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$? I want to parametrise so I can use the divergence theorem to calculate the flux along the surface above. I don't know how to do it and would like ...
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1answer
31 views

Is there a method to parameterise any surface? And how could I parametrise this one given?

I'm having major trouble every time I need to parametrise a surface in order to take a surface integral, I just have no idea where to even start half of the time. Is there some kind of method that can ...
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0answers
74 views

Tweaking Reddit's Ranking Algorithm

This image explains how Reddit's Ranking algorithm works. As you know, Reddit is a very high traffic site. Therefore, the post rank decreases quite fast. This algorithm puts emphasis on bringing ...
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3answers
184 views

Find the coordinates of the point where the normal cuts the curve again problem

Find the equation of the normal to the curve $x=2\cos\theta$, $y=3\sin\theta$ at the point where $\theta=\frac{1}{4}\pi$. Find the coordinates of the point where this normal cuts the curve again. ...
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2answers
30 views

To find a extremal point of a function with parameters

I have a function $$f(x) = (x-5m)(x+m)^2$$ I have tried to find the extremal points of this function (and then find if it's local maxima or minima). That means I need to find the x of derivative. The ...
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0answers
31 views

Finding Extrema of a Function Restrained by a Parametrized Surface

Where on the parametrized surface $r(u,v)=⟨u^2,v^3,uv⟩$ is the temperature $T(x,y,z)=12x+y−12z$ minimal? Find all local maxima, local minima or saddle points. I know that one has to insert $r(u,v)$ ...
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1answer
32 views

Parametric equation for a circle centered at $(-5,-4)$ with a radius of $4$: Why is $t$ negative?

The parametric equations for $x$ & $y$ are as follows: $$x=-5+4 \cos (-t)$$ $$y=-4+4 \sin (-t)$$ My question is: Why is $t$ negative in this case? Thanks for any help.
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28 views

What is the implicit form of $x=cos(t),y=-3+cos(2t)$? [closed]

I know I have to use the properties of the trigonometric functions but I don't know which of them would help me get the answer.
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1answer
58 views

Nurbs parametric coordinate span

I am using the Nurbs definition of Wikipedia. I might have missed something in the definition but I cannot understand how to know on which interval does the parametric coordinate span. Particularily ...
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2answers
32 views

Find point(s) of intersection between a line and a circle whose radius is parameterized by the same variable as the line

Let's assume we have a line: $$\begin{align} x&: x_0 + v_xt, \\ y&: y_0 + v_yt \end{align}$$ and a circle $$\begin{align} x&: X_0 + kt\cos(s), \\ y&: Y_0 + kt\sin(s).\end{align}$$ ...
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3answers
39 views

How a formula is developed

The rule for converting line equations to parametric equations is: $$\frac{(x-x_1)}{a} =\frac{(y-y_1)}{b} =t$$ I would like to know how this was developed. Thank you.
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1answer
43 views

Find the end points of a line segment in 3D space

I have a line segment in 3 dimensional space (x,y,z), and I want to find the 2 endpoints of this line segment. Is there a systematic way of doing this? To be specific, I have the line described by ...
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1answer
49 views

Finding twice-differentiable function of x of a parametric curve when dx/dt = 0

We're working on finding tangents of parametric curves and I feel like this problem isn't as hard as I'm making it out to be, but I am completely stumped. I am given this information: Given $x=3cos(...
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1answer
44 views

Finding an equation to the surface S that is bounded between $z=x^2-y^2$ inside the cylinder $x^2+y^2=1$

How to find a parametric equation to the surface S that is bounded between $z=x^2-y^2$ inside the cylinder $x^2+y^2=1$, and while C be the the Boundary of that surface. While reading the solution of ...
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1answer
456 views

Equation of a quadric surface on which this curve lies?

I am currently learning about surfaces. So for the parametrized curve: $r=\langle t^2, 3t\cos(2t), 3t\sin(2t)\rangle,\quad t\ge 1$ how can I find a equation for the surface the curve lie? Also what ...
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0answers
63 views

Cartesian/Parametric 3d equation of a cheese twist?

Hi I'm looking for the equation of a cheese twist in 3d (either parametric or cartesian)... Can be multiple planes but was wondering if anyone had any idea to execute something like this? Thanks e.g.
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4answers
195 views

Parametric to implicit form of a curve

"Find the implicit form of the curve defined by parametric equations $x = t+1,y=\frac{1}{t^{2}}$" How can I clear $t$ to arrive at the implicit equation?
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1answer
25 views

Parametric equations of a line

"Find the parametric equations of a line that passes through point $(1,1,0)$, parallel to plane $2x+3y+z=7$ and perpendicular to the line $\frac{x-1}{-2}= \frac{y}{3}=-z-2$" I don't know where to ...
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1answer
284 views

How to convert the parametric equation into implicit form?

This problem is generated from another Green's theorem related question of mine. The original equation of the plane curve is not in rational parametric form. In order to calculate the symbolic ...
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1answer
279 views

Point on surface where tangent plane is perpendicular to line.

I'm given the surface $ x^3-2y^2+z^2=27 $ and have to find where the tangent plane is perpendicular to the line described by \begin{align*} x &= 3t-5 \\ y &= 2t+7\\z&=1-t\sqrt2\end{align*} ...
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1answer
70 views

Conversion between trig functions and hyperbolic trig functions

Using trig identities we can see that $\sin^2 x + \cos^2 x = \tanh^2 x + \text{sech}^2 x = 1$ , and so the parametric graph $(\cos t, \sin t)$ is similar to $(\text{sech} t, \tanh t)$. The first ...
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1answer
33 views

Slope of a Parametrized Curve

Say that we have the parametrized curve $x=e^{3t}, y=te^{-t}$. What would be the slope of this at the point $(1,0)$ and also on which points on the curve would the curve be horizontal? What I have ...
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2answers
47 views

Given two curves, find parametric curve

I am given two graphs x versus t and y versus t and I have to determine the parametric curve. The two graphs I am given: Parametric curve (that is the right answer): So the solutions say that: ...
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0answers
45 views

Is it possible to turn the parametric equation of a line in 3 dimensions into the general equation?

I Know it is impossible to do so since the parametric equation for a plane is the intersection of $2$ planes.For example: $x$ $=$ $\frac{-5}{4t}+\frac{1}{4}$; $y=\frac{3}{4t}+\frac{5}{4}$; $z=t$ ...
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1answer
95 views

Surface integral: Cone cut by a cylinder

Ok I've got this exercise from Apostol I'm trying to do: "The cylinder $x²+y²=2x$ cuts out a portion of a surface S from the upper nappe of the cone x²+y²=z². Compute the value of the integral: $$\int\...
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1answer
46 views

Parametric equations of perpendicular lines

I'm having problems with this: Find the parametric equation of the line that passes through the point $(-1, 4, 5)$ and is perpendicular to the line: $$x = -2 + t$$ $$y = 1 - t$$ $$z = 1 + 2t$$
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1answer
269 views

Parametric equations - locus at midpoint

Consider the parametric equations $x=-2t^2$ and $y=4t$ The normal at any point, P, cuts the x-axis at Q. Find the Cartesian equation of the locus of the midpoint, M, of PQ. Can anyone help get me ...
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1answer
39 views

Counting the integer soultions to this parametric inequality

hello I am looking for an efficient way, hopefully a formula or a somewhat tight upper bound, for the number of integer solutions to the following let $k$ be a fixed integer and $\lambda \ge 1$ and $...