Question: If \|z – 2 – i\| = \|z\| sin $\left( \frac{\pi}{4} - \arg z \right)$ then locus of z is/ are...

Question

If |z – 2 – i| = |z| sin $\left( \frac{\pi}{4} - \arg z \right)$ then locus of z is/ are

Answer 1

Sol. |(x – 2) + i (y – 1)| = |z| $\left( \frac{1}{\sqrt{2}}\cos\theta - \frac{1}{\sqrt{2}}\sin\theta \right)$

q = arg z

Ž $\sqrt{(x - 2)^{2} + (y - 1)^{2}}$

= $\frac{1}{\sqrt{2}}$ |z| cos q – $\frac{1}{\sqrt{2}}$ |z| sin q

= $\frac{1}{\sqrt{2}}$ (x – y)

Ž parabola with vertex (2, 1)

directrix x – y = 0

Question: If \|z – 2 – i\| = \|z\| sin \(\left( \frac{\pi}{4} - \arg z \right)\) then locus of z is/ are...