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.
