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

detailed solution

Correct option is C

Let z=x+iy Polar form of a complex number z=r(cos⁡θ+isin⁡θ) Where r=|z| and θ=arg⁡(z)⇒x+t˙y=r(cos⁡θ+isin⁡θ)⇒x=rcos⁡θ,y=rsin⁡θ Given |z−2−i|=|z|sin⁡π4−arg⁡z⇒|x+iy−2−i|=rsin⁡π4−θ⇒|(x−2)+i(y−1)|=r12cos⁡θ−12sin⁡θ⇒(x−2)2+(y−1)2=12rcos⁡θ−12rsin⁡θ⇒(x−2)2+(y−1)2=12(x−y)∴ Locus of z is a parabola with focus (2,1) and directrix x−y=0 ( ∵ Locus of a moving point ' P′ in a plane which such that the distance  From the point ' P′ to fixed point is equal to distance from the point ' P′ to fixed line is  parabola)