TS EAMCET · Maths · Complex Number
Let \(z=x+i y\) represent a point \(\mathrm{P}(x, y)\) in the Argand plane. If \(z\) satisfies the condition that amplitude of \(\frac{z-3}{z-2 i}=-\frac{\pi}{2}\), then the locus of P is
- A the circle \(x^2+y^2-3 x-2 y=0\)
- B the arc of the circle \(x^2+y^2-3 x-2 y=0\) intercepted by the diameter \(2 x+3 y-6=0\) containing the origin and excluding the points \((3,0)\) and \((0,2)\)
- C the arc of the circle \(x^2+y^2-3 x-2 y=0\) intercepted by the diameter \(2 x+3 y-6=0\) not containing the origin and excluding the points \((3,0)\) and \((0,2)\)
- D the circle \(x^2+y^2-3 x-2 y=0\) not containing the point \((0,2)\)
Answer & Solution
Correct Answer
(B) the arc of the circle \(x^2+y^2-3 x-2 y=0\) intercepted by the diameter \(2 x+3 y-6=0\) containing the origin and excluding the points \((3,0)\) and \((0,2)\)
Step-by-step Solution
Detailed explanation
\(\frac{z-3}{z-2i} = \frac{((x-3)+iy)(x-i(y-2))}{x^2+(y-2)^2} = \frac{(x^2+y^2-3x-2y) + i(2x+3y-6)}{x^2+(y-2)^2}\)…
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