QUESTION

# If $a + ib = c + id$ , then $\left( A \right)a - c = i\left( {b - d} \right) \\ \left( B \right)a - ib = c - id \\ \left( C \right)a = d,b = c \\ \left( D \right){\text{None of these}} \\$

Hint: In this question, we use the concept of equal complex numbers. A complex number is a number that can be expressed in the form $x + iy$ , where x and y are real numbers. If two complex numbers are equal that means their real and imaginary parts are equal.

Now, we have two complex numbers $a + ib$ and $c + id$ both are equal to each other.
First complex number, $a + ib$ in which a is real part and b is imaginary part.
Second complex number, $c + id$ in which c is real part and d is imaginary part.
Now, both complex numbers are equal; that means their real and imaginary parts are equal; that is called equal complex numbers.
$\Rightarrow a + ib = c + id$
After comparing, real and imaginary parts are equal.
$\Rightarrow a = c{\text{ and }}b = d$
Now,
$\Rightarrow a - c = id - ib \\ \Rightarrow a - c = i\left( {d - b} \right) \\$
As we know b=d,
$\Rightarrow a - c = i\left( {b - d} \right)$
Hence, option (A) is correct.
Now,
$\Rightarrow a + ib = c + id \\ \Rightarrow a - id = c - ib \\$
As we know b=d,
$\Rightarrow a - ib = c - id$
Hence, option (B) is correct.
So, the correct options are (A) and (B).

Note: In such types of problems we use some important points to solve questions in an easy way. First we have to equal both complex numbers and find the relation between their real and imaginary parts and then check the following options by using this relation.