Mathematics Stack Exchange
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Questions tagged [parametric]

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

In mathematics, a parametric equation of a curve is a representation of this curve through equations expressing the coordinates of the points of the curve as functions of a variable called a parameter. This contrasts with implicit equations that define a curve as the zero set of some equation in the coordinates.

The parametric forms of curves are well-suited for drawing on a computer, while their corresponding implicit forms are useful for analytic manipulations (intersections, etc.)

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Parametric equations; two objects intersects

Two objects move through the air. The movement of the first can be described by the equation $(4t, 6+3t)$, the second by $(20-70(t-2)\cos u, 70(t-2)\sin u)$, where the parameter $t$ represents time and $u$ is the angle between the ground and the…
Random Noob
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Relation between an equation coefficients and equations's solutions

Let consider a second degree $$a(k)x^2+b(k)x+c(k)=0$$ Suppose I want to find conditions on $k$ so that, called $x_1$ and $x_2$ the two solutions of the equation, they satisfy the following $$\frac{1}{x_1^2} +\frac{1}{x_2^2}=\frac{1}{4}$$ Could you…
Maryam
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Intuitive way to eliminate a parameter

I need to eliminate $\theta$ from the equations $x=\sin\theta+\cos\theta$ and $y=\tan\theta+\cot\theta$. I am actually provide with a hint: consider $x^2y$ , which worked nicely for me. However I am wondering if it was possible to "see" this…
user71207
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Why is there a point on the unit circle that is not represented by these parametric equations?

$$x=\frac{2t+1}{2t^{2}+2t+1}$$ $$y=\frac{2t^{2}+2t}{2t^{2}+2t+1}$$By squaring $x$ and $y$ and adding them up, I obtained $x^2 + y^2 = 1$ after some algebraic manipulation. But the question asks which point is not represented by these parametric…
user71207
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Parametric equation gives dy/dt = sin(t) +3 and x(t)= 6t^2+ln(t) and asks for when the graph of the position will have a horizontal tangent

To my understanding if $\dfrac{dy}{dt}=0$ then the tangent is horizontal, but $\sin(t)+3$ will never equal $0$, therefore there is no horizontal tangent. Thanks!
user10262038
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How do I find the value of k based on the fact that x+k is a tangent to a parametric equation?

I have been given the question of: A curve has the parametric equations x=2$t^2$ and y=4t. Find the value(s) of k if y=x+k is a tangent to the curve. Being the first question I've gotten of this nature I have no idea on how to go about it.
loubloom
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How would you solve this equation by graphing? Or any other method?

A particle is moving so that its position at time $t$ is given by the parametric equations \begin{align*} x &= 5\sin(-2t) \\ y &= 5\cos(2t). \end{align*}What is the speed of the particle? A particle's position in the plane $t$ seconds after it…
user834359
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What is my next step in finding the ordered pairs for the tangent plane

The solution i have envisioned is that I need to find the point of tangency and the vector that defined as the crossproduct of $R_x$ $R_y$ to be able to solve for the tangent plane where $u=u_0$ and $v=v_0$. after obtaining a tangent plane equation…
Ross Henderson
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Finding the parametric equation of a curve

The problem is to find the parametric equation of the line that is tangent to the line of intersection between the plane $x+2y+3z=6$ and the surface $x^2+y^2=2$ and passes through the point $(1,1,1)$. How would I solve this problem?
minnn
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How would I go about parameterizing the "unit square"?

I was looking at the equation |x|+|y|=1 and an interesting thought came to my mind. Could you parameterize this equation by angle into something akin to a square sine and square cosine? How exactly would you go about doing this?
O.S.
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For which of the following values of real number $t$, the equation $x^4-tx+\dfrac 1t = 0$ has no root on the interval $[1,2]$.

For which of the following values of real number $t$, the equation $x^4-tx+\dfrac 1t = 0$ has no root on the interval $[1,2]$. I think we might check for which $t$ $y=tx-\dfrac 1t $ have solution on [1,16].
piteer
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Point on line closest to origin given the line's parametric equations

So I'm given the line described by: $$\begin{cases} x = \frac{2}{3} + \lambda \\ y = \frac{1}{3} + \lambda \\ z = \lambda \end{cases}$$ And I'm asked to determine the point that is closest to the origin. I have the following formula for the distance…
Matthias K.
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system with parameters

So our teacher gave us some things to work on, he said we should try to find out how to solve these tasks, but I'm the worst at math, I tried to think of something and failed. I'd appreciate some help! Thank you in advance!
Mel
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How would I express the following parametric pair in Cartesian form? $\bigl(4\sin(4t), 3\sin(3t)\bigr)$

Since I know that $\sin(4t) = 4\cos^3t\sin t-4\cos t\sin^3t$ and $\sin(3t) = 3\sin t-4\sin^3 t$ (or at least I think I do), I think I'm halfway there.
Austen Clark
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Problem Solving This Question?

A parametric curve is described by the following equations $\dfrac{\text{d}x}{\text{d}t} =x$, $y=\cos(t)$, $z=\sin(t)$, and passes through ⟨1, 1, 0⟩ when $t = 0$. By solving the ODE for $x(t)$, or otherwise, find an expression for $x$ in terms of…
user467597
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