WBJEE · Maths · Complex Number
The value of \(|z|^{2}+|z-3|^{2}+|z-i|^{2}\) is minimum when \(z\) equals
- A \(2-\frac{2}{3} i\)
- B \(45+3 i\)
- C \(1+\frac{i}{3}\)
- D \(1-\frac{i}{3}\)
Answer & Solution
Correct Answer
(C) \(1+\frac{i}{3}\)
Step-by-step Solution
Detailed explanation
Let \(z=x+i y\) \(\therefore|z|^{2}+|z-3|^{2}+|z-i|^{2}\) \[ \begin{aligned} =|x+i y|^{2}+\mid &\mid x-3)+\left.i y\right|^{2} \\ &+|x+i(y-1)|^{2} \end{aligned} \] \(=x^{2}+y^{2}+(x-3)^{2}+y^{2}+x^{2}+(y-1)^{2}\) \(=x^{2}+y^{2}+x^{2}-6 x+9+y^{2}+x^{2}+y^{2}\)…
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