If a^2 + b^2 = 1, then 1 + b + ia 1 + b - ia = - Vidyadip

MCQ

If ${a^2} + {b^2} = 1,$ then $\frac{{1 + b + ia}}{{1 + b - ia}} = $

A
$1$
B
$2$
✓
$b + ia$
D
$a + ib$

✓

Answer

Correct option: C.

$b + ia$

c
(c) Given that ${a^2} + {b^2} = 1$, therefore
$\frac{{1 + b + ia}}{{1 + b - ia}} = \frac{{(1 + b + ia)(1 + b + ia)}}{{(1 + b - ia)(1 + b + ia)}}$
$ = \frac{{{{(1 + b)}^2} - {a^2} + 2ia(1 + b)}}{{1 + {b^2} + 2b + {a^2}}}$$ = \frac{{(1 - {a^2}) + 2b + {b^2} + 2ia(1 + b)}}{{2(1 + b)}}$
$ = \frac{{2{b^2} + 2b + 2ia(1 + b)}}{{2\,(1 + b)}} = b + ia$
Trick : Put $a = 0,b = 1$, $\frac{{1 + b + ia}}{{1 + b - ia}} = \frac{{1 + 1 + 0}}{{1 + 1 - 0}} = 1$
But options $(a)$ and $ (c)$ give $1.$
So again put $a = 1,b = 0,\frac{{1 + b + ia}}{{1 + b - ia}} = \frac{{1 + i}}{{1 - i}} = i$.
Which gives $ (c)$ only.

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