If z is a complex number satisfying the condition |z−2−i|=|z|sinπ4−arg z , then locus of z is

Let Z=x+iyPolar form of a complex number Z=r(cos⁡θ+isin⁡θ)where r=|Z| and θ=arg⁡(z)⇒x+iy=r(cos⁡θ+isin⁡θ)⇒x=rcos⁡θ,y=rsin⁡θGiven |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)⇒parabola with focus (2,1) directrix x - y = 0( Locus of a moving point ‘P’ in a plain 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.)

If z is a complex number satisfying the condition |z−2−i|=|z|sinπ4−arg z , then locus of z is

Let Z=x+iyPolar form of a complex number Z=r(cos⁡θ+isin⁡θ)where r=|Z| and θ=arg⁡(z)⇒x+iy=r(cos⁡θ+isin⁡θ)⇒x=rcos⁡θ,y=rsin⁡θGiven |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)⇒parabola with focus (2,1) directrix x - y = 0( Locus of a moving point ‘P’ in a plain 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.)

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If z is a complex number satisfying the condition $| z - 2 - i | = | z | \sin (\frac{π}{4} - \arg z)$ , then locus of z is

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Pair of straight lines

Ellipse

Circle

Parabola

answer is C.

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Detailed Solution

Let $Z = x + iy$
Polar form of a complex number $Z = r (\cos θ + isin θ)$
where $r = | Z | and θ = \arg (z)$
$\Rightarrow x + iy = r (\cos θ + isin θ) \Rightarrow x = rcos θ, y = rsin θ$
Given $| z - 2 - i | = | z | \sin (\frac{π}{4} - \arg z)$
$\begin{array}{l} \Rightarrow | x + iy - 2 - i | = rsin (\frac{π}{4} - θ) \\ \Rightarrow | (x - 2) + i (y - 1) | = r (\frac{1}{\sqrt{2}} \cos θ - \frac{1}{\sqrt{2}} \sin θ) \\ \Rightarrow \sqrt{(x - 2)^{2} + (y - 1)^{2}} = \frac{1}{\sqrt{2}} rcos θ - \frac{1}{\sqrt{2}} rsin θ \\ \Rightarrow \sqrt{(x - 2)^{2} + (y - 1)^{2}} = \frac{1}{\sqrt{2}} (x - y) \end{array}$
$\Rightarrow$ parabola with focus (2,1) directrix x - y = 0
( Locus of a moving point ‘P’ in a plain 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.)