Ellipse Compression and Expansion

In summary, ellipse compression and expansion is a phenomenon where the shape of an ellipse changes due to a change in its parameters, such as its major and minor axes. This change is caused by a change in the eccentricity of the ellipse, and does not affect the area of the ellipse. Ellipses can undergo compression and expansion in any direction, and this phenomenon is commonly observed in nature and used in engineering and design.
  • #1
cocoavi
11
0
An ellipse is defined by the equation [tex] \frac {x^2}{144} + \frac {y^2}{36} = 1[/tex]. Determine the equation of the ellipse formed when the original ellipse has undergone a horizontal compression by a factor of 1/2 and a vertical expansion by a factor of 3.

I'm not quite sure how to work out the equation.. could someone give me some hints?
 
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  • #2
You need to recognize the standard form for the equation of an ellipse:
[tex] \frac {x^2}{a^2} + \frac {y^2}{b^2} = 1[/tex]

That's the equation for an ellipse centered at the origin. The length of the horizontal axis is 2a; the vertical axis is 2b.
 
  • #3


To determine the new equation of the ellipse after compression and expansion, we can use the general equation for an ellipse:

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

where (h,k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes, respectively.

First, let's identify the center of the original ellipse. We can see that the equation is in the form \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, which means the center is at (0,0).

Next, we can find the values of a and b by taking the square root of the denominators in the equation. In this case, a = 12 and b = 6.

Now, for the horizontal compression by a factor of 1/2, we need to divide the length of the semi-major axis by 2. This means that the new value of a is 6.

Similarly, for the vertical expansion by a factor of 3, we need to multiply the length of the semi-minor axis by 3. This means that the new value of b is 18.

Therefore, the new equation of the ellipse is:

\frac{x^2}{36} + \frac{y^2}{324} = 1

We can also write this in the general form as:

\frac{(x-0)^2}{6^2} + \frac{(y-0)^2}{18^2} = 1

which matches the general equation we started with, confirming that our calculations are correct.

In summary, when an ellipse is horizontally compressed by a factor of 1/2 and vertically expanded by a factor of 3, the new equation is:

\frac{(x-h)^2}{(\frac{a}{2})^2} + \frac{(y-k)^2}{(3b)^2} = 1

where (h,k) is the center of the original ellipse and a and b are the lengths of the semi-major and semi-minor axes, respectively.
 

Related to Ellipse Compression and Expansion

1. What is ellipse compression and expansion?

Ellipse compression and expansion is a phenomenon where the shape of an ellipse changes due to a change in its parameters, such as its major and minor axes.

2. What causes ellipse compression and expansion?

Ellipse compression and expansion is caused by a change in the eccentricity of the ellipse. When the eccentricity increases, the ellipse becomes more elongated and experiences compression. When the eccentricity decreases, the ellipse becomes more circular and experiences expansion.

3. How does ellipse compression and expansion affect the area of an ellipse?

Ellipse compression and expansion do not affect the area of an ellipse. The area of an ellipse is determined by its major and minor axes, not its eccentricity. Therefore, as long as the major and minor axes remain constant, the area of an ellipse will remain the same.

4. Can an ellipse only undergo compression and expansion in one direction?

No, an ellipse can undergo compression and expansion in any direction. This is because the shape of an ellipse is symmetrical, so any change in its parameters will affect the entire ellipse.

5. Is ellipse compression and expansion a common occurrence in nature?

Yes, ellipse compression and expansion can be observed in many natural phenomena, such as the orbits of planets and moons, the shape of galaxies, and the motion of comets. It is also commonly used in engineering and design, such as in the construction of bridges and arches.

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