Ellipse Compression and Expansion

[cocoavi]

An ellipse is defined by the equation $$\frac {x^2}{144} + \frac {y^2}{36} = 1$$. 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?

 

[Doc Al]

You need to recognize the standard form for the equation of an ellipse:
$$\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1$$

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

 

[thepatientmental]

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.