Mathematics - Complex Number Question with Solution | TestHub

Given,

$z_1=2+3i$ and $z_2=3+4i$ and the set

$\mathrm{S}=\left\{\mathrm{z}\in \mathrm{C}:{\left|\mathrm{z}-{\mathrm{z}}_1\right|}^2-{\left|\mathrm{z}-{\mathrm{z}}_2\right|}^2={\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|}^2\right\}$,

Now putting the value of $z=x+iy, z_1=2+3i$ and $z_2=3+4i$ in ${\left|\mathrm{z}-{\mathrm{z}}_1\right|}^2-{\left|\mathrm{z}-{\mathrm{z}}_2\right|}^2={\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|}^2$ we get,

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

$\Rightarrow x+y=7$

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