• # question_answer If ${{z}_{1}}=10+6i,{{z}_{2}}=4+6i$ and $z$ is a complex number such that $amp\left( \frac{z-{{z}_{1}}}{z-{{z}_{2}}} \right)=\frac{\pi }{4},$ then the value of $|z-7-9i|$ is equal to [IIT 1990] A) $\sqrt{2}$ B) $2\sqrt{2}$ C) $3\sqrt{2}$ D) $2\sqrt{3}$

Given numbers are ${{z}_{1}}=10+6i,{{z}_{2}}=4+6i$and $z=x+iy$ \ $amp\left( \frac{z-{{z}_{1}}}{z-{{z}_{2}}} \right)=\frac{\pi }{4}$Þ $amp\left[ \frac{(x-10)+i\,(y-6)}{(x-4)+i\,(y-6)} \right]=\frac{\pi }{4}$ Þ $\frac{(x-4)(y-6)-(y-6)(x-10)}{(x-4)(x-10)+{{(y-6)}^{2}}}=1$ Þ $12y-{{y}^{2}}-72+6y={{x}^{2}}-14x+40$     .....(i) Now  $|z-7-9i|\,=|\,(x-7)+i(y-9)|$ Þ $\sqrt{{{(x-7)}^{2}}+{{(y-9)}^{2}}}$                 ....(ii) From (i),  $({{x}^{2}}-14x+49)+({{y}^{2}}-18y+81)=18$ Þ ${{(x-7)}^{2}}+{{(y-9)}^{2}}=18$ or  ${{[{{(x-7)}^{2}}+{{(y-9)}^{2}}]}^{1/2}}={{[18]}^{1/2}}=3\sqrt{2}$ \ $|(x-7)+i(y-9)|=3\sqrt{2}$or$|z-7-9i|=3\sqrt{2}$.