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Let z_{1}=2+3 i and z_{2}=3+4 i.The set S=\left\{z \in C:\left|z-z_{1}\right|^{2}-\left|z-z_{2}\right|^{2}=\left|z_{1}-z_{2}\right|^{2}\right\}represents a

Option: 1

hyperbola with the length of the transverse axis 7


Option: 2

hyperbola with eccentricity 2


Option: 3

straight line with the sum of its intercepts on the coordinate axes equals −18


Option: 4

straight line with the sum of its intercepts on the coordinate axes equals 14


Answers (1)

best_answer

Let \quad \mathrm{z}=\mathrm{x}+\mathrm{iy}

\mathrm{z}-\mathrm{z}_{1}=(\mathrm{x}-2)+\mathrm{i}(\mathrm{y}-3)
\left|\mathrm{z}-\mathrm{z}_{1}\right|^{2}=(\mathrm{x}-2)^{2}+(\mathrm{y}-3)^{2}
\mathrm{z}-\mathrm{z}_{2}=(\mathrm{x}-3)+\mathrm{i}(\mathrm{y}-4)
\left|\mathrm{z}-\mathrm{z}_{2}\right|^{2}=(\mathrm{x}-3)^{2}+(\mathrm{y}-4)^{2}
\left((x-2)^{2}+(y-3)^{2}\right)-\left((x-3)^{2}+(y-4)^{2}\right)=2

\Rightarrow 2 \mathrm{x}+2 \mathrm{y}=14
=\mathrm{x}+\mathrm{y}=7

straight line with sum of intercept on C.A = 14

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rishi.raj

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