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UPSC NDA I 2016 Mathematics Solved Paper

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For the next two (2) items that follow

Let z be a complex number satisfying

\left|\frac{z-4}{z-8}\right|=1

and

\left|\frac{z}{z-2}\right|=\frac{3}{2}

[NDA I 2016]

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Question : 52 of 120

Marks: +1, -0

What is |z| equal to ?

6
12
18
36

Solution:

\left|\frac{z-4}{z-8}\right|=1

and

\left|\frac{z}{z-2}\right|=\frac{3}{2}

\Rightarrow\left|z-4\right|=\left|z-8\right|

Let

z=x+iy

\left|x+iy-4\right|=\left|x+iy-8\right|

Squaring both sides, we get

\left[(x-4)^{2}+y^{2}\right]=\left[(x-8)^{2}+y^{2}\right]

(x-4)^{2}=(x-8)^{2}

\Rightarrow x^{2}+16-8x=x^{2}+64-16x

\Rightarrow 8x=48 \Rightarrow x=6

when

\left|\frac{z}{z-2}\right|=\frac{3}{2}

\Rightarrow 2\left|z\right|=3\left|z-2\right|

Squaring both sides, we get

4(x^{2}+y^{2})=9\left[(x-2)^{2}+y^{2}\right]

\Rightarrow 4x^{2}+4y^{2}=9x^{2}+36-36x+9y^{2}

\Rightarrow 5x^{2}+5y^{2}-36x+36=0

as we know

x=6

5(6)^{2}+5y^{2}-36\times 6+36=0

\Rightarrow 5y^{2}=0 \Rightarrow y=0

Hence

x=6

and

y=0

\Rightarrow z=6

\left|z\right|=6

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