Area of an ellipse.
By Emma Valentine •
I need to find the area of the image of a circle centred at the origin with radius 3 under the transformation:
$ \begin{pmatrix} 3 & 0\\ 0 & \frac{1}{3} \end{pmatrix} $
The image is the ellipse $ \frac{x^2}{81}+y^2=1$. It would appear that it has the same area as the original circle i.e. $9\pi$. Is this because the matrix has some special property such as being its own inverse?
$\endgroup$ 52 Answers
$\begingroup$Yes, this matrix has a special property, namely its determinant is 1.
$\endgroup$ 2 $\begingroup$It is a known formula that the area enclosed in the ellipse with semi-axes $a$ and $b$ is $\pi ab$, as may be seen from the orthogonal affinity that transforms a circle into an ellipse.
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