A) \[\sqrt{2}\]
B) \[2\sqrt{2}\]
C) \[3\sqrt{2}\]
D) \[2\sqrt{3}\]
Correct Answer: C
Solution :
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\[|z-7-9i|=3\sqrt{2}\].
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