Mathematics - Complex Number Question with Solution | TestHub

Note that $\left|z-3\sqrt{2}\right|$ is the distance between the point $z$ and $\left(3\sqrt{2},0\right)$ on argand plane.

Note that $\left|z-p\sqrt{2}i\right|$ is the distance between the point $z$ and $\left(0,p\sqrt{2}\right)$ on argand plane.

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$\left|z-3\sqrt{2}\right|+\left|z-p\sqrt{2}i\right|$ is sum of distance of $z$ from $A\left(3\sqrt{2},0\right)$ and $B\left(0,p\sqrt{2}\right)$

For minimising the sum, $z$ should lie on $AB$ and $AB$ should be equal to $5\sqrt{2}$

Now, ${\left(AB\right)}^2=18+2p^2$

$\Rightarrow 18+2p^2={\left(5\sqrt{2}\right)}^2$

$\Rightarrow p^2=16$

i.e. $p=\pm 4$