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Question: The locus represented by the complex equation \|z – 2 – i\| = \|z\| sin \(\left( \frac{\pi}{4} - \ar...

The locus represented by the complex equation |z – 2 – i| = |z| sin (π4argz)\left( \frac{\pi}{4} - \arg z \right)is the part of –

A

A pair of straight lines

B

A circle

C

A parabola

D

A rectangular hyperbola

Answer

A parabola

Explanation

Solution

Sol. Let z = x + iy = r (cos q + i sin q), then the equation is

| (x – 2) + i(y – 1)| = r (12cosθ12sinθ)\left( \frac{1}{\sqrt{2}}\cos\theta - \frac{1}{\sqrt{2}}\sin\theta \right)

= 12\frac{1}{\sqrt{2}} (r cos q – r sin q)

or (x2)2+(y1)2\sqrt{(x - 2)^{2} + (y - 1)^{2}} = 12\frac{1}{\sqrt{2}} (x – y).

which is the part of a parabola with focus (2, 1) and directrix x – y = 0.