Question
The locus of a point in the Argand plane that moves satisfying the equation $$\left| {z - 1 + i} \right| - \left| {z - 2 - i} \right| = 3:$$
A.
is a circle with radius $$3$$ and center at $$z = \frac{3}{2}$$
B.
is an ellipse with its foci at $$1 – i$$ and $$2 + i$$ and major axis $$= 3$$
C.
is a hyperbola with its foci at $$1 – i$$ and $$2 + i$$ and its transverse axis $$= 3$$
D.
None of the above
Answer :
is a hyperbola with its foci at $$1 – i$$ and $$2 + i$$ and its transverse axis $$= 3$$
Solution :
The given eq. implies that the difference between the distances of the moving point from two fixed points $$\left( {1 - i} \right)$$ and $$\left( {2 + i} \right)$$ is constant using the property of the hyperbola that the difference between the focal distances of any point on the curve is constant, the locus in reference is therefore a hyperbola.