KCET · Maths · Complex Number
If \(P(x, y)\) denotes \(z=x+i y\) in Argand's plane and \(\left|\frac{z-1}{z+2 i}\right|=1\), then the locus of \(P\) is a/an
- A hyperbola
- B ellipse
- C circle
- D straight line
Answer & Solution
Correct Answer
(D) straight line
Step-by-step Solution
Detailed explanation
Given, \(z=x+i y\) and \(\left|\frac{z-1}{z+2 i}\right|=1\)
\(\Rightarrow \quad\left|\frac{(x+i y)-1}{(x+i y)+2 i}\right|=1\)
\(\Rightarrow \quad|(x-1)+i y|=|x+(y+2) i|\)
Squaring on both sides,
\(\Rightarrow \quad|(x-1)+i y|^{2}=|x+(y+2) i|^{2}\) \(\Rightarrow \quad(x-1)^{2}+y^{2}=x^{2}+(y+2)^{2}\) \(\Rightarrow \quad x^{2}+y^{2}+1-2 x=x^{2}+y^{2}+4+4 y\) \(\Rightarrow 2 x+24+3=0\), which represents a straight line.
\(\Rightarrow \quad\left|\frac{(x+i y)-1}{(x+i y)+2 i}\right|=1\)
\(\Rightarrow \quad|(x-1)+i y|=|x+(y+2) i|\)
Squaring on both sides,
\(\Rightarrow \quad|(x-1)+i y|^{2}=|x+(y+2) i|^{2}\) \(\Rightarrow \quad(x-1)^{2}+y^{2}=x^{2}+(y+2)^{2}\) \(\Rightarrow \quad x^{2}+y^{2}+1-2 x=x^{2}+y^{2}+4+4 y\) \(\Rightarrow 2 x+24+3=0\), which represents a straight line.
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