For z ∈ ℂ if the minimum value of ( |z - 3 2 | + |z - p 2 i|) is 5...

Note that $\left|z - 3\sqrt{2}\right|$ is the distance between the point $z and \left(3\sqrt{2}, 0\right) on argand plane.</p><p>Note that \left|z - p\sqrt{2}i\right| is the distance between the point$z and \left(0, p\sqrt{2}\right)$on argand plane.</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/ba6c7191-3d0e-4094-b2a0-b7d1f2ccdec3-image.png" style="width: 257px; height: 140px;" /></p><p>$\left|z - 3\sqrt{2}\right| + \left|z - p\sqrt{2}i\right| is sum$of distance ofz from$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 + 2{p}^{2}$</p><p>$\to 18 + 2{p}^{2} = {\left(5\sqrt{2}\right)}^{2}

$\to {p}^{2} = 16$

i.e. $p = ± 4$