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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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Help me to sketch this parametric curves

Is there any defined process to sketch parametric curves? Thanks in advance. $$x = \cos^2 t, \quad y = 1 - \sin t, \quad 0 \leq t \leq 2\pi.$$
Diego Pacheco
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For what values do these parametric equations draw a circle in the clockwise direction?

So I've got this set of parametric equation on the x-y plane $x=cos(ln(5t))$ $y=sin(ln(5t))$ for $t>0$. I need to find the range of values for which the circle is drawn in the clockwise direction. Initially I tried for values where $t>0.2$ and…
tsp216
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Steps to solve parametric equation of Folium of Descartes

I know that the parametric equations of the Folium of Descartes are $\displaystyle x = \frac{3t}{(1+t)^3} $and $\displaystyle y = \frac{3t^2}{(1+t)^3}$. What are the steps to achieve this parametric equation from the given equation $x^3 + y^3 =…
Sue
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Parametric Curves finding its cartesian equation

A curve has parametric equations: $x=2\csc(X)$, $y=\cot(X)$. How do I find the cartesian equation of the curve? Thanks in advance.
James
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How do I find the 2 slopes at which this parametric function crosses itself?

I have a parametric function. If you graph it, you'll find that it looks like a figure 8. x(t) = 2sin(2t) y(t) = 8sin(t) How do I find the slopes of the function when it intersects with itself again (graph this to see what I mean)? Should I convert…
Minden Petrofsky
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How to find the parametric equation

I've got this curve and I know its part of a circle, From this I should find that $x = a-acos(\theta) $and $y = a-asin(\theta)$ but I don't know how? Can someone help?
Ayoub Rossi
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Conversion of parametric equation to Cartesian form

Can anyone help me convert this equation from parametric form to its Cartesian form? $$ x = 7(\sec (t) + \tan (t)) $$ $$ y = 7(\sec (t) - \tan(t)) $$
shmop
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Trajectory of $X_1(t) = (2\cos^2 x, \cos t \sin t)\;\; (0\le t \le \pi)$

I need to draw trajectory of $X_1(t) = (2\cos^2 t, \cos t \sin t)\;\; (0\le t \le \pi)$ If one let $2\cos^2 x = x, \cos t \sin t =y$ then $dx/dt = -4\cos t\sin t =-4y$ and one could derive x = -4yt+C. However, with this approach doesn't make the t…
Beverlie
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Finding whether a point lies inside or outside a curve on the parametric domain

I have a parametric surface and a curve in the parametric domain of that surface (let's say u and v), and findally a point somewhere on that surface at parameters u and v. How do I determine if that point is within the curve or not?
Red
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How to express the surface $ \ y^{2}+z^{2}=15 \ $ between $x=-6$ and $x=8$ in parametric form?

How to express the surface $ \ y^{2}+z^{2}=15 \ $ between $x=-6$ and $x=8$ in parametric form? From $ y^{2}+z^{2}=15 $ , we can have $$ y=\sqrt {15} \cos t \quad \text{ and }\quad z=\sqrt{15} \sin t,$$ but how to manage $x=-6$ to $x=8$? …
MAS
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Cylindrical surface equation of parametric curve

Given the helix curve: $$r(t) = 5\sin(t) \,\mathbf i + 12t\,\mathbf j - 5\cos(t)\,\mathbf k$$ What is the equation of the circular cylindrical surface which contains the helix curve and is parallel to the y-axis? I think the answer to this problem…
user3567004
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Arc parameter equation

So, I have the function $y=-x^2+3$, and I'm to generate the parametric equation for the arc that is above $y=x$. I got from $-x^2+3>x$ to $\frac{1-\sqrt{13}}{-2}
Grak
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Show that the chord of contact of tangents from the point $(a,-a)$ to the parabola $x^2=4ay$ has length $5a$

Show that the chord of contact of tangents from the point $(a,-a)$ to the parabola $x^2=4ay$ has length $5a$. My current method uses the distance formula and takes super long, is there some easier method that i'm missing? Thanks in advance!
kjhg
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Parametric equation reorganization

I'm not very familiar with parametric equations and have come across this textbook problem: Find a system of two equations in three variables $x_1$, $x_2$, and $x_3$, that has the solution set given by the parametric representation $x_1 = t$, $x_2…
mikeglaz
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Find the equation of the chord of contact AB of tangents drawn from an external point $(X1,Y1)$ to the parabola $X^2=-36y$

Q: Find the equation of the chord of contact AB of tangents drawn from an external point $(X1,Y1)$ to the parabola $X^2=-36y$ My working: Let $A(2aA,aA^2), B(2aB, aB^2)$ -> Equation of chord AB $ y=\frac{A+B}{2}x-aAB$ -> Point of tangent…
kjhg
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