TS EAMCET · Maths · Complex Number
\(z=x+i y\) and the point \(P\) represents \(z\) in the Argand plane. If the amplitude of \(\left(\frac{2 z-i}{z+2 i}\right)\) is \(\frac{\pi}{4}\), then the equation of the locus of P is
- A \(2 x^2+2 y^2-3 x+3 y-2=0,(x, y) \neq(0,-2)\)
- B \(2 x^2+2 y^2+5 x+3 y-2=0,(x, y) \neq(0,-2)\)
- C \(2 x^2+2 y^2+3 x+3 y-2=0,(x, y) \neq(0,2)\)
- D \(2 x^2+2 y^2-5 x+3 y-2=0,(x, y) \neq(0,2)\)
Answer & Solution
Correct Answer
(B) \(2 x^2+2 y^2+5 x+3 y-2=0,(x, y) \neq(0,-2)\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { } z=x+i y \\ & \begin{array}{l}\left(\frac{2 z-i}{z+2 i}\right)=\frac{2 x+2 i y-i}{x+i y+2 i}=\frac{2 x+i(2 y-1)}{x+(y+2) i} \\ =\frac{(2 x+i(2 y-1))(x-(y+2) i)}{x^2+(y+2)^2} \\ =\frac{2 x^2+(2 y-1)(y+2)+i((2 y-1) x-2 x(y+2))}{x^2+(y+2)^2} \\…
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