# How to calculate parametric representation of a circle?

• geft
In summary, the conversation discusses solving for coordinates that satisfy a given equation, specifically y^2 + 4y + z^2 = 5 in the y-z plane. The equation is rewritten in the form of a circle and then parametrized to find the correct answer. The conversation also touches on finding tangent vectors and unit tangent vectors for a given curve.
geft

## Homework Statement

$$y^2 + 4y + z^2 = 5, x = 3$$

## The Attempt at a Solution

I know that the calculated coordinates must satisfy the above equation, but I don't know how to go about solving for those coordinates. The best I could do was to equate $$z = \sqrt{(-y + 5)(y + 1)}$$.

Hmm,
$$y^2 + 4y + z^2 = 5$$
doesn't really look like a circle, does it?
I mean, the general equation for a circle (in the y-z plane) is
$$(y - a)^2 + (z - b)^2 = r^2$$

Can you maybe start by rewriting it to this form?

For the next step, remember the usual parametrisation for a circle $y^2 + z^2 = r^2$ is
$$y(t) = r \sin t, z(t) = r \cos t$$

Oh! By completing the square, I get

$$(y + 2)^2 + (z)^2 = 3^2$$

$$r(t) = (3, 3 \sin t, 3 \cos t)$$

Is this correct?

Last edited:
Almost!
The first line is right. Just think very carefully about what you call sin(t).
When in doubt, calculate
$$(3 \sin t)^2 + 4 (3 \sin t) + (3 \cos t)^2$$
and check if it gives 5.

Is it $$r(t) = (3, 3 \sin t - 2, 3 \cos t)$$?

Why isn't it 3 sin t + 2? I had to change the sign to get the answer to fit.

Very well done.

The sign is there, because you want 3 (y + 2/3) to be 3 cos(t).

Thank you very much. I have another (somewhat) related question, but I'd rather not create a new thread. I certainly hope you don't mind.

## Homework Statement

Given a curve C: r(t), find a tangent vector r'(t), a unit tangent vector u'(t), and the tangent of C at P.

$$r(t) = (\cosh t, \sinh t), P: (\frac{5}{3}, \frac{4}{3})$$

## The Attempt at a Solution

$$r'(t) = (\sinh t, \cosh t)$$
$$u'(t) = (\cosh 2x)^{-1/2}(\sinh t, \cosh t)$$

The last part stumps me.

You mean about u'(t)?
It is simply a unit vector along r'(t), so you will have to divide r'(t) by its length.
To get the cosh(2x) there is just some (hyperbolic) identities magic using the formulas analagous to $$\cos^2 t + \sin^2 t = 1$$ and $$\cos^2 t - \sin^2 t = \sin 2t$$.

## 1. How do you calculate the center and radius of a circle using parametric representation?

The center of a circle (h,k) can be found by taking the average of the x and y coordinates of the circle's parametric equations, which are given by x = r*cos(t) and y = r*sin(t). The radius (r) can be calculated by taking the square root of the sum of the squares of the coefficients of the x and y terms, which is equivalent to the Pythagorean theorem.

## 2. What is the purpose of using parametric representation to describe a circle?

Parametric representation allows for a more flexible and versatile way of describing a circle, compared to the standard cartesian coordinate form (x^2 + y^2 = r^2). It allows us to express the circle in terms of a third variable (t), which can be manipulated to create different shapes and orientations of the circle.

## 3. How do you convert a standard circle equation into parametric representation?

To convert a standard circle equation (x^2 + y^2 = r^2) into parametric representation, we can use the following equations: x = r*cos(t) and y = r*sin(t). These equations describe the x and y coordinates of the circle in terms of a variable t, which ranges from 0 to 2π.

## 4. Can parametric representation be used to describe circles in higher dimensions?

Yes, parametric representation can be extended to describe circles in higher dimensions. For example, in three dimensions, the parametric equations for a circle would be x = r*cos(t), y = r*sin(t), and z = 0. These equations can also be altered to create circles in non-planar surfaces.

## 5. How do you graph a circle using parametric representation?

To graph a circle using parametric representation, we first need to choose a range of values for t (usually 0 to 2π). Then, we can plug in these values into the parametric equations x = r*cos(t) and y = r*sin(t) to find the corresponding x and y coordinates. Plotting these coordinates on a graph will create the circle.

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