Transformation Sending a Plane to a Line

If the determinant of a matrix is zero, the image of  
  after transformation by this matrix will not be the whole of  
. It will generally be a line.
Consider the transformation of  
  represented by the matrix  
\[M= \left| \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right|\]
. Let a general point in  
\[\left( \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right) \begin{pmatrix}a\\b\end{pmatrix}=\begin{pmatrix}a+2b\\a+2b\end{pmatrix}\]
The result of the transformation,  
  and the who;e  
  is sent to the line  

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