For a complex number a such that 0 < |a| < 1, which of the following statements is true?  

Concept:

The argument of a can be any real number, as it represents the angle that the line connecting the origin to the point    makes with the positive real axis.

Explanation:

Given,  and    is a complex number.

We are to find which statement is true.

Option 1: This implies that for a complex number , if  is less than 1, the modulus of   is less than . To verify

if this is true, we can try specific examples for z and a under the condition that |z| < 1 and .

Let  and  First, check the moduli:

This satisfies the condition  and |z| < 1 .

Clearly,  , so the inequality does not hold for this example.

The statement is false because we found a counterexample where || is not less than |z - a| .

Option2: This implies that if the modulus of  is equal to 1 - , then  = 1 . This looks like

a symmetry condition related to distances in the complex plane. For points on the unit circle (i.e., |z| = 1 ),

this is true because the modulus of z would satisfy this equation. Therefore, this statement is true.

Option 3: This implies that for a point on the unit circle (i.e., |z| = 1 ), the modulus of z - a is less than |1 -  | .

To verify this, we will try specific examples for z and a under the condition that |z| = 1 and  .

Let z = 1 (since |z| = 1 ) and  

Since z = 1 and  (because a is real), 

So the inequality  does not hold in this example because both sides are equal.

Therefore, the statement is not true.

Option 4: This suggests that if the modulus of 1 -  is less than , then z is inside the unit circle. 

Let   and  .

Since  , we have  (because  is real).

In this case,   and .

Clearly,  , so the condition  does not hold for this example,

and hence it does not verify the statement.

The statement seems to not hold.

The  option 2) is correct.