Linear Transformation - Circle to ellipse

[Javé]

Find a linear transformation of the plane that deforms a unit circle centered at the origin in a major half-axis ellipse measuring on a straight y = kx and minor half axis with measure equal to b.

 

[HallsofIvy]

I'm not sure I understand what "measuring on a straight y= kx" means.
I will assume it means that one axis lies on the line y= kx and that the other is perpendicular to it, y= (-1/k)x.
Since this is on a plane the linear transformation can be represented as a 2 by 2 matrix, $\displaystyle \begin{bmatrix}a & b \\ c & d \end{bmatrix}$. If the x-axis is transformed to y= (-1/k)x and has length b, (1, 0) is mapped to $\displaystyle \left(\frac{kb}{\sqrt{k^2+ 1}},\frac{-b}{\sqrt{k^2+1}}\right)$. In order to complete this I think you will need to be given the length of the major semi-axis as well.