TS EAMCET · Maths · Complex Number
If a complex number \(z=x+i y\) represents a point \(\mathrm{P}(x, y)\) in the Argand plane and \(z\) satisfies the condition that the imaginary part of \(\frac{z-3}{z+3 i}\) is zero, then the locus of the point P is
- A \(x^2+y^2-3 x+3 y=0,(x, y) \neq(0,-3)\)
- B \(2 x y-3 x+3 y+9=0,(x, y) \neq(0,-3)\)
- C \(x-y-3=0,(x, y) \neq(0,-3)\)
- D \(x+y+3=0,(x, y) \neq(0,-3)\)
Answer & Solution
Correct Answer
(C) \(x-y-3=0,(x, y) \neq(0,-3)\)
Step-by-step Solution
Detailed explanation
\(Im\left(\frac{z-3}{z+3 i}\right) = 0\) \(Im\left(\frac{(x-3)+iy}{x+i(y+3)}\right) = 0\) \(Im\left(\frac{((x-3)+iy)(x-i(y+3))}{(x+i(y+3))(x-i(y+3))}\right) = 0\) \(Im\left(\frac{x(x-3) - i(x-3)(y+3) + ixy - i^2y(y+3)}{x^2 + (y+3)^2}\right) = 0\)…
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