Simply complex!

This may look like a brainful at first, but it's actually pretty easy to solve. Remember that a diameter has endpoints (extremities) that are on the circle . This is the key, because it means that both endpoints ( $z_1$ and $z_2$ ) will satisfy the equation. Therefore, I plugged in $z_1$ into the equation to get: ${|z_1 - z_1|}^2 + {|z_1 - z_2|}^2 = k$ Simplify some: ${|0|}^2 + {|z_1 - z_2|}^2 = k$ $0^2 + {|z_1 - z_2|}^2 = k$ $0 + {|z_1 - z_2|}^2 = k$ ${|z_1 - z_2|}^2 = k$ Now substitute the values of $z_1$ and $z_2$ in: ${|2 + 3i - (4 + 3i)|}^2 = k$ Simplify some: ${|2 + 3i - 4 - 3i)|}^2 = k$ ${|-2|}^2 = k$ $2^2 = k$ $4 = k$ Now use the Symmetric Property of Equality: $k = 4$

Let $P(z)$ be a point on the circle with $A(z_1) ~ \text{and} ~ B(z_2)$ as extremities of a diameter . Using the property that angle in a semicircle is $\dfrac{ \pi} {2}$ ,

$AP^2 + BP^2 = AB^2 \text{ ( Pythagoras' theorem) }$ $\implies | z- z_1|^2 + |z- z_2 |^2 = | z_1 - z_2 |^2 = | 2+ 3i - 4- 3i |^2 = 4$

$\therefore \color{#20A900}{ k = 4 }$