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Question

If |z2i|=|z|sin(π4arg(z)), where z is in first quadrant, then the locus of z is

A
a ray.
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B
a part of a circle.
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C
a part of a parabola.
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D
a part of an ellipse.
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Solution

The correct option is C a part of a parabola.
Given : |z2i|=|z|sin(π4arg(z))

Let z=x+iy and arg(z)=θtanθ=yx, so
|x2+i(y1)|=|z|sin(π4arg(z))(x2)2+(y1)2=x2+y2×|xy|2x2+y2(x2)2+(y1)2=|xy|2

This represents a parabola, whose focus is (2,1) and equation of directrix is y=x

Hence, the locus of z is a part of a parabola.

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