1993 Paper 3 Q10

Year: 1993
Paper: 3
Question Number: 10

Course: LFM Pure
Section: Linear transformations

Difficulty: 1700.0 Banger: 1484.0

linear transformations matrix parallelogram coordinate geometry area preservation determinant parallel lines

Problem

The transformation $T$ of the point $P$ in the $x$,$y$ plane to the point $P'$ is constructed as follows: \hfil\break Lines are drawn through $P$ parallel to the lines $y=mx$ and $y=-mx$ to cut the line $y=kx$ at $Q$ and $R$ respectively, $m$ and $k$ being given constants. $P'$ is the fourth vertex of the parallelogram $PQP'R$. Show that if $P$ is $(x_1,y_1)$ then $Q$ is $$ \left( {mx_1-y_1 \over m-k}, {k(mx_1-y_1)\over m-k}\right). $$ Obtain the coordinates of $P'$ in terms of $x_1$, $y_1$, $m$ and $k$, and express $T$ as a matrix transformation. Show that areas are transformed under $T$ into areas of the same magnitude.

No solution available for this problem.

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Difficulty Rating: 1700.0

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Banger Rating: 1484.0

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Problem source

The transformation $T$ of the point $P$ in the $x$,$y$ plane to
the point $P'$ is constructed as follows:
\hfil\break
Lines are drawn through $P$ parallel to the lines $y=mx$ and $y=-mx$
to cut the line $y=kx$ at $Q$ and $R$ respectively, $m$ and $k$ being
given constants. $P'$ is the fourth vertex of the parallelogram
$PQP'R$. 
Show that if $P$ is $(x_1,y_1)$ then $Q$ is 
$$
\left( {mx_1-y_1 \over m-k}, {k(mx_1-y_1)\over m-k}\right).
$$
Obtain the coordinates of $P'$ in terms of 
$x_1$, $y_1$, $m$ and $k$, and express $T$ as a matrix transformation.
Show that  areas are transformed under $T$ into areas of the same 
magnitude.

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