Let `z_(1)` and `z_(2)` be the roots of the equation `z^(2)+pz+q=0`. Suppose `z_(1)` and `z_(2)` are represented by points A and B in the Argand plane such that `angleAOB=alpha`, where O is the origin.
Statement-1: If OA=OB, then `p^(2)=4q cos^(2)alpha/2`
Statement-2: If affix of a point P in the Argand plane is z, then `ze^(ia)` is represented by a point Q such that `anglePOQ =alpha` and `OP=OQ`.

Options

A. Statement-1 is True, Statement-2 is True: Statement-2 is a correct exp,anation for statement-1.
B. Statement-1 is true, statement -2 is true, Statement-2 is not a correct explanation for statement-1.
C. Statement-1 is True, statement-2 is false,
D. statement-1 is False, Statement-2 is true.

Answer

To solve the problem, we need to analyze the given statements and derive the required equation step by step.

### Given:
- The roots of the equation \( z^2 + pz + q = 0 \) are \( z_1 \) and \( z_2 \).
- The roots can be represented as points \( A \) and \( B \) in the Argand plane, with \( \angle AOB = \alpha \).
- We need to prove that if \( OA = OB \), then \( p^2 = \frac{4q \cos^2(\alpha/2)}{} \).

### Step 1: Represent the roots in polar form
Let:
- \( z_1 = r e^{i\theta} \)
- \( z_2 = r e^{i(\theta + \alpha)} \)

Here, \( r \) is the modulus (length of OA and OB), and \( \theta \) is the argument of \( z_1 \).

### Step 2: Find the sum and product of the roots
From Vieta's formulas:
- The sum of the roots \( z_1 + z_2 = -p \)
- The product of the roots \( z_1 z_2 = q \)

Calculating the sum:
\[ z_1 + z_2 = r e^{i\theta} + r e^{i(\theta + \alpha)} = r(e^{i\theta} + e^{i(\theta + \alpha)}) \]
Using the formula for the sum of complex exponentials:
\[ z_1 + z_2 = r \left( e^{i\theta} + e^{i\theta} e^{i\alpha} \right) = r e^{i\theta} (1 + e^{i\alpha}) \]

### Step 3: Calculate the modulus of the sum
The modulus of \( z_1 + z_2 \) can be expressed as:
\[ |z_1 + z_2| = r |1 + e^{i\alpha}| \]
Using the formula for the modulus:
\[ |1 + e^{i\alpha}| = \sqrt{(1 + \cos\alpha)^2 + \sin^2\alpha} = \sqrt{2(1 + \cos\alpha)} = 2 \cos(\alpha/2) \]
Thus,
\[ |z_1 + z_2| = r \cdot 2 \cos(\alpha/2) \]

### Step 4: Relate it to \( p \)
Since \( z_1 + z_2 = -p \), we have:
\[ |-p| = |z_1 + z_2| = r \cdot 2 \cos(\alpha/2) \]
This implies:
\[ p = 2r \cos(\alpha/2) \]

### Step 5: Calculate the product of the roots
Now, calculate the product:
\[ z_1 z_2 = r e^{i\theta} \cdot r e^{i(\theta + \alpha)} = r^2 e^{i(2\theta + \alpha)} = q \]
Thus,
\[ |z_1 z_2| = r^2 \]
This implies:
\[ q = r^2 \]

### Step 6: Substitute into the equation
Now we have:
\[ p^2 = (2r \cos(\alpha/2))^2 = 4r^2 \cos^2(\alpha/2) \]
Since \( q = r^2 \), we can substitute to get:
\[ p^2 = 4q \cos^2(\alpha/2) \]

### Conclusion
We have shown that if \( OA = OB \), then:
\[ p^2 = 4q \cos^2(\alpha/2) \]