Transforming the Ellipse

Algebra Level 5

Given an ellipse (blue) with its semi-minor axis along the vector \( (2, 1) \), and its semi-major axis along the vector \( (-2, 4) \). The ellipse is transformed by multiplying the position vector of each of its points by a matrix \( A \), which is given by

\[ A = \begin{bmatrix} 0.5 & .75 \\ -0.25 & 1 \end{bmatrix} \]

The resulting curve is another ellipse (red). Find the length of semi-major axis of the transformed ellipse. (To three decimal places)

This problem is original.

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