# The Attempt at a SolutionReduce Ellipse: Centre & Eccentricity

• zorro
In summary, to reduce an ellipse, you need to find the distance between its center and one of its foci, known as the semi-major axis, and divide it by the eccentricity of the ellipse to get the semi-minor axis. The center of an ellipse is the point at the intersection of its major and minor axes, equidistant from all points on the ellipse. Eccentricity is a measure of how "oval-shaped" an ellipse is, calculated by dividing the distance between the center and one of the foci by the length of the semi-major axis. To find the center of an ellipse, you need to know the coordinates of at least three points on the ellipse and use a mathematical formula. You can reduce an ellipse
zorro

## Homework Statement

Reduce the given ellipse in standard form and find its centre and eccentricity.

4(x-2y+1)2 + 9(2x+y+2)2 = 25

## Homework Equations

Rotation of axes
x=Xcosθ - Ysinθ
y=Xsinθ + Ycosθ

I would not use the "rotation of axes" formulas. Instead, let X= x- 2y and Y= 2x+ y so that your equation becomes
$$\frac{4}{25}(X- (-1))^2+ \frac{9}{25}(Y- (-2))^2= 1$$

What is the center and eccentricity of that ellipse? Now go back to x and y coordinates.

Wow! that is a very nice method. Thanks!

## 1. How do you reduce an ellipse?

To reduce an ellipse, you need to find the distance between its center and one of its foci. This distance is known as the semi-major axis. Then, divide the semi-major axis by the eccentricity of the ellipse. The result is the semi-minor axis of the reduced ellipse.

## 2. What is the center of an ellipse?

The center of an ellipse is the point that lies at the intersection of its major and minor axes. It is equidistant from all points on the ellipse.

## 3. What is eccentricity in an ellipse?

Eccentricity is a measure of how "oval-shaped" an ellipse is. It is represented by the letter e and is calculated by dividing the distance between the center and one of the foci by the length of the semi-major axis.

## 4. How do you find the center of an ellipse?

To find the center of an ellipse, you need to know the coordinates of at least three points on the ellipse. Then, you can use a mathematical formula to calculate the center point.

## 5. Can you reduce an ellipse with any center and eccentricity?

Yes, you can reduce an ellipse with any center and eccentricity. However, the resulting reduced ellipse may not have the same shape as the original ellipse. The center and eccentricity will determine the size and orientation of the reduced ellipse.

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