11th Class Mathematics Complex Numbers and Quadratic Equations Question Bank Critical Thinking

  • 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) \[2\sqrt{3}\]

    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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