MCQ
If $z = (\lambda + 3) + i\sqrt {5 - {\lambda ^2},} $ then the locus of $z$ is a
- ✓Circle
- BStraight line
- CParabola
- DNone of these
✓
Answer
Correct option: A.
Circle
a
(a) Let $z = x + iy$.Then $x = \lambda + 3$and $y = \sqrt {5 - {\lambda ^2}} $
==> ${(x - 3)^2} = {\lambda ^2}$ ......$(i)$
and ${y^2} = 5 - {\lambda ^2}$ ......$(ii)$
From $(i) $ and $(ii)$, ${(x - 3)^2} = 5 - {y^2}$==> ${(x - 3)^2} + {y^2} = 5$.
Obviously it is a circle.
(a) Let $z = x + iy$.Then $x = \lambda + 3$and $y = \sqrt {5 - {\lambda ^2}} $
==> ${(x - 3)^2} = {\lambda ^2}$ ......$(i)$
and ${y^2} = 5 - {\lambda ^2}$ ......$(ii)$
From $(i) $ and $(ii)$, ${(x - 3)^2} = 5 - {y^2}$==> ${(x - 3)^2} + {y^2} = 5$.
Obviously it is a circle.
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