# Changing Center of Ellipse in Polar Coordinates

• alpha25
In summary, there is an easy way to change the center of a circle or ellipse in polar coordinates. It involves transforming the equation into a specific form and redefining certain constants. This method is commonly found in textbooks and can be applied to any ellipse, not just special cases.
alpha25
Hi, does exist an easy way to change the center of circle or a ellipse in polar coordinates?

thanks!

Yes, and it was given in any textbook dealing with the conic sections that I have ever seen:
If the equation of an ellipse centered at (0, 0) is
$$\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1$$
then the same ellipse, centered at (a, b) has equation
$$\frac{(x- a)^2}{a^2}+ \frac{(y-b)^2}{b^2}= 1$$

Now you have to pass from the cartesian eqns that Halls wrote to polar coordinates and you're done.

Yes thanks...but I need it in polar coordinates

Insert polar representations for "x" and "y", multiply out parentheses and simplify and redefine variables/constants.
In particular, remember simplifying trig identities, such as, for example:
$$2\sin^{2}\theta=1-\cos(2\theta)$$

1 person
Now, HallsofIvy made out a special case, with the center with the same values as the lengths of the semi-axes.

You shouldn't make that restriction here (call one of the (a,b)-pairs (c,d)-for example).

To give you the first step on your way, multiplying up and out, we get (with (c,d) centre coordinates):
$$b^{2}r^{2} \cos^{2}\theta+a^{2}r^{2} \sin^{2}\theta-2b^{2}cr \cos\theta-2a^{2} dr\sin\theta=a^{2}b^{2}-c^{2}-d^{2}$$
There would be various ways to simplify this expression further, and redefing independent constants.

One very compact way of doing so would be to transform your equation into the following form:
$$Ar^{2}\cos\gamma+Br\sin\phi=C$$
where the angle "phi" is a phase-shifted version of "gamma"/2 with a fourth constant D to be determined along with A, B and C (gamma being twice the value of "theta")

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## 1. What is the formula for finding the center of an ellipse in polar coordinates?

The formula for finding the center of an ellipse in polar coordinates is (a, b), where a represents the distance from the origin to the center of the ellipse and b represents the distance from the origin to the focus of the ellipse.

## 2. How does changing the center of an ellipse affect its shape in polar coordinates?

Changing the center of an ellipse in polar coordinates will shift the entire ellipse, but it will not change its shape. The distance from the origin to the focus and the distance from the origin to any point on the ellipse will remain the same, thus preserving the shape of the ellipse.

## 3. Can the center of an ellipse in polar coordinates be negative?

Yes, the center of an ellipse in polar coordinates can be negative. This means that the center of the ellipse is located in the opposite direction from the positive axis. It does not affect the shape or size of the ellipse, only its position.

## 4. How do you graph an ellipse with a center at (0,0) in polar coordinates?

To graph an ellipse with a center at (0,0) in polar coordinates, you will need to plot the focus at (0,b) and then plot points along the ellipse using the formula r = a(1-e^2)/(1+e*cos(theta)), where a is the length of the semi-major axis and e is the eccentricity of the ellipse.

## 5. How can you determine the center of an ellipse in polar coordinates given its equation?

To determine the center of an ellipse in polar coordinates given its equation, you can rewrite the equation in standard form and then compare it to the general form of an ellipse in polar coordinates, which is r = a(1-e^2)/(1+e*cos(theta)). The values of a and b in the standard form will correspond to the coordinates of the center in the general form.

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