Question

If $z=(3+7i)(p+iq)$, where $p,q\in I-\left\{0\right\}$, is a purely imaginary, then minimum value of ${\left|z\right|}^2$ is

• A. $0$
• B. $58$
• C. $\frac{3364}{3}$
• D. $3364$

Answer 1

The correct option is D $3364$

$z=(3+7i)(p+iq)\dots \left(1\right)$

$z=(3p-7q)+i(3q+7p)$

Now, $z$ is purely imaginary

$3p-7q=0$

or $\frac{p}{q}=\frac{7}{3}$

$\frac{p}{q}+i=\frac{7}{3}+i$

Therefore, $\frac{(p+qi)}{q}=\frac{7+3i}{3}$

$p+iq=7+3i\dots \left(2\right)$

Substitute $\left(2\right)$ in $\left(1\right)$

$z=21+9i+49i-21$

Thus, $z=58i$

$\left|z^2\right|=3364$