Chapter 11.7, Problem 24HP

To determine

To find:Which pair does not represent a dilation with center at the Origin.

Answer to Problem 24HP

Option-C does not represent Dilation.

Explanation of Solution

The dilation is $\left(x,y\right)\to \left(kx,ky\right)$

Where $k$ is scale factor.

Taking Option-A

Given

  $\begin{array}{l}A=\left(2,3\right)\\ A\text{'}=\left(4,6\right)\end{array}$

  $\begin{array}{l}A\left(2,3\right)\to A\text{'}\left(k\left(2\right),k\left(3\right)\right)\\ \Rightarrow A\left(2,3\right)\to A\text{'}\left(2k,3k\right)\end{array}$

Given $A\text{'}=\left(4,6\right)$

On comparing

We get

  $\begin{array}{l}2k=4\\ 3k=6\end{array}$

On solving both of them

We get

  $\begin{array}{l}2k=4\\ \Rightarrow \frac{2k}{2}=\frac{4}{2}\\ \Rightarrow k=2\\ 3k=6\\ \Rightarrow \frac{3k}{3}=\frac{6}{3}\\ \Rightarrow k=2\end{array}$

In both cases we got $k=2$

So, Option A represents Dilation.

Taking Option-B

Given

  $\begin{array}{l}B=\left(4,6\right)\\ B\text{'}=\left(2,3\right)\end{array}$

  $\begin{array}{l}B\left(4,6\right)\to B\text{'}\left(k\left(4\right),k\left(6\right)\right)\\ \Rightarrow B\left(4,6\right)\to B\text{'}\left(4k,6k\right)\end{array}$

Given $B\text{'}=\left(2,3\right)$

On comparing

We get

  $\begin{array}{l}4k=2\\ 6k=3\end{array}$

On solving both of them

We get

  $\begin{array}{l}4k=2\\ \Rightarrow \frac{4k}{4}=\frac{2}{4}\\ \Rightarrow k=\frac{1}{2}\\ 6k=3\\ \Rightarrow \frac{6k}{6}=\frac{3}{6}\\ \Rightarrow k=\frac{1}{2}\end{array}$

In both cases we got $k=\frac{1}{2}$

So, Option B represents Dilation.

Taking Option-C

Given

  $\begin{array}{l}C=\left(2,5\right)\\ C\text{'}=\left(4,7\right)\end{array}$

  $\begin{array}{l}C\left(2,5\right)\to C\text{'}\left(k\left(2\right),k\left(5\right)\right)\\ \Rightarrow C\left(4,6\right)\to C\text{'}\left(2k,5k\right)\end{array}$

Given $C\text{'}=\left(4,7\right)$

On comparing

We get

  $\begin{array}{l}2k=4\\ 5k=7\end{array}$

On solving both of them

We get

  $\begin{array}{l}2k=4\\ \Rightarrow \frac{2k}{2}=\frac{4}{2}\\ \Rightarrow k=2\\ 5k=7\\ \Rightarrow \frac{5k}{5}=\frac{7}{5}\\ \Rightarrow k=\frac{7}{5}\end{array}$

In both cases we got different $k$ values.

So, Option Cdoes not represent Dilation.

Taking Option-D

Given

  $\begin{array}{l}D=\left(-2,4\right)\\ D\text{'}=\left(-1,2\right)\end{array}$

  $\begin{array}{l}D\left(-2,4\right)\to D\text{'}\left(k\left(-2\right),k\left(4\right)\right)\\ \Rightarrow D\left(-2,4\right)\to D\text{'}\left(-2k,4k\right)\end{array}$

Given $D\text{'}=\left(-1,2\right)$

On comparing

We get

  $\begin{array}{l}-2k=-1\\ 4k=2\end{array}$

On solving both of them

We get

  $\begin{array}{l}4k=2\\ \Rightarrow \frac{4k}{4}=\frac{2}{4}\\ \Rightarrow k=\frac{1}{2}\\ -2k=-1\\ \Rightarrow \frac{-2k}{-2}=\frac{-1}{-1}\\ \Rightarrow k=\frac{1}{2}\end{array}$

In both cases we got $k=\frac{1}{2}$

So, Option D represents Dilation.

Therefore,

We can say that

Option-C does not represent Dilation.