9 A transformation T acts on all points in the plane. The image of a general point P is denoted by \(\mathrm { P } ^ { \prime }\). \(\mathrm { P } ^ { \prime }\) always lies on the line \(y = 2 x\) and has the same \(y\)-coordinate as P. This is illustrated in Fig. 9.
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\caption{Fig. 9}
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- Write down the image of the point \(( 10,50 )\) under transformation T .
- P has coordinates \(( x , y )\). State the coordinates of \(\mathrm { P } ^ { \prime }\).
- All points on a particular line \(l\) are mapped onto the point \(( 3,6 )\). Write down the equation of the line \(l\).
- In part (iii), the whole of the line \(l\) was mapped by T onto a single point. There are an infinite number of lines which have this property under T. Describe these lines.
- For a different set of lines, the transformation T has the same effect as translation parallel to the \(x\)-axis. Describe this set of lines.
- Find the \(2 \times 2\) matrix which represents the transformation.
- Show that this matrix is singular. Relate this result to the transformation.