Question

3) Find a and b if
i) (a + ib) (1 + i) = 2 + i​

Answer 1

$\large\bold{\underline{\underline{Answer:-}}}$

$\bold{ \to(a + ib) (1 + i) = 2 + i }\\ \\ \bold{\to a + ai + ib + b{(i)}^{2} = 2 + 1} \\ \\ \bold{\to</p><p>a + ai + ib + b (-1) = 2 + i }\\ \\ \bold{\to</p><p>a + ai + ib - b = 2 + i }\\ \\ \bold{ \to </p><p>(a - b) + (a + b)i = 2 + i}</p><p>$

$\underline \bold{ Compare \: imaginary \: and \: real \: part \: </p><p> \: \: }$

$\to \bold{a - b = 2 --------- (i) } \\ \bold{ \to</p><p>a + b = 1 ------- (ii) }</p><p>$

$\underline\bold{Add \: equation \: (i) \: and \: (ii) \: \: \: }$

$\to \bold{(a - b) + (a + b) = 2 + 1} \\ \\ \to\bold{2a = 3} \\ \\ \to\bold{ a = \frac{3}{2} }$

$\underline \bold{Put \: the \: value \: of \: a = \frac{3}{2} \: in \: equation \: (ii)}$

$\to \bold{a + b = 1} \\ \\ \to \bold{ \frac{3}{2} + b = 1 }\\ \\ \to \bold{b = 1 - \frac{3}{2} } \\ \\ \to \bold{b = \frac{2 - 3}{2}} \\ \\ \to \bold{b = \frac{ - 1}{2} }$

$\bold{Hence \: , } \: </p><p> \: \fbox{ \bold{ a = \frac{3}{2} }}$ $\bold{ and } \: \: \fbox{b = \bold{ \frac{ - 1}{2} }}$

Answer 2

$\huge\bf\underline\red{Solution}$

$\bf\underline\purple{Need to know}$

i² = -1

(a + ib) ( 1 + i ) = 2 + i

a ( 1 + i ) + ib ( 1 + i) = 2+i

a + ai +ib + i²b = 2 + i

a + ai +ib (-1)b = 2 + i

a - b + ( a + b ) i = 2 + 1 ( i)

Comparing both sides Imaginary and Real

a - b = 2

a + b = 1

_________

2a = 3

__________

a = 3/2

Sub value of a in any two equations

From 1

3/2 - b =2

3/2 -2 = b

b = -1/2

So values of a,b are 3/2, -1/2