Let z _ 1 = 2 + 3i and z _ 2 = 3 + 4i . The set S = \ z ∈ C: |z - z...

Given,

${z}_{1} = 2 + 3i$ and ${z}_{2} = 3 + 4i$ and the set

$S = \left\{z \in C: {\left|z - {z}_{1}\right|}^{2} - {\left|z - {z}_{2}\right|}^{2} = {\left|{z}_{1} - {z}_{2}\right|}^{2}\right\}$,

Now putting the value of $z = x + iy, {z}_{1} = 2 + 3i$ and ${z}_{2} = 3 + 4i$ in ${\left|z - {z}_{1}\right|}^{2} - {\left|z - {z}_{2}\right|}^{2} = {\left|{z}_{1} - {z}_{2}\right|}^{2}$ we get,

$⇒ \left(( x - 2{)}^{2} + ( y - 3{)}^{2}\right) - \left(( x - 3{)}^{2} - ( y - 4{)}^{2}\right) = 1 + 1$

$⇒ x + y = 7$

Which is a straight line with intercept $7$ on $x \text{ and } y$ axis, so sum of intercept will be $7 + 7 = 14$.