Question

If ${\mathrm{z}}_1=\left(4,5\right)$ and ${\mathrm{z}}_2=\left(-3,2\right)$then $\frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}$equals.

• A. $\left(\frac{-23}{12},\frac{-2}{13}\right)$
• B. $\left(\frac{2}{13},\frac{23}{13}\right)$
• C. $\left(\frac{-2}{13},\frac{-23}{13}\right)$
• D. $\left(\frac{-2}{13},\frac{23}{13}\right)$

Answer 1

The correct option is C

$\left(\frac{-2}{13},\frac{-23}{13}\right)$

Explanation for the correct option:

Finding the value of $\frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}$:

The given points are ${\mathrm{z}}_1=\left(4,5\right)$ and ${\mathrm{z}}_2=\left(-3,2\right)$

Write the points in complex equation ,

${\mathrm{z}}_1=4+5\mathrm{i}$and ${\mathrm{z}}_2=-3+2\mathrm{i}$,

Now,

$\begin{array}{c}\frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}=\frac{4+5\mathrm{i}}{-3+2\mathrm{i}}\times \frac{-3-2\mathrm{i}}{-3-2\mathrm{i}}\left[\mathrm{By}\mathrm{conjugate}\right]\\ =\frac{\left(4+5\mathrm{i}\right)\left(-3-2\mathrm{i}\right)}{\left(-3+2\mathrm{i}\right)\left(-3-2\mathrm{i}\right)}\\ =\frac{-12-8\mathrm{i}-15\mathrm{i}-10{\mathrm{i}}^2}{{\left(-3\right)}^2-{\left(2\mathrm{i}\right)}^2}\\ =\frac{-12-23\mathrm{i}-10\left(-1\right)}{9-4\left(-1\right)}\left[\because i^2=-1\right]\\ =\frac{-2-23\mathrm{i}}{13}\\ =-\frac{2}{13}-\frac{23}{13}\mathrm{i}\end{array}$

$\therefore \frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}=\left(\frac{-2}{13},\frac{-23}{13}\right)$

Hence, the correct option is C.