If `z^(2)+(p+iq)z+(r+is)=0`, where,p,q,r,s are non-zero, has real roots, then

Options

A. `pqs=s^(2)+q^(2)r`
B. `pqr=r^(2)+p^(2)s`
C. `prs=q^(2)+r^(2)p`
D. `qrs=p^(2)+s^(2)q`

Answer

To solve the equation \( z^2 + (p + iq)z + (r + is) = 0 \) for real roots, we will follow these steps:

### Step 1: Identify the coefficients
The coefficients of the quadratic equation are:
- \( a = 1 \)
- \( b = p + iq \)
- \( c = r + is \)

### Step 2: Use the quadratic formula
The roots of the quadratic equation can be found using the quadratic formula:
\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ z = \frac{-(p + iq) \pm \sqrt{(p + iq)^2 - 4 \cdot 1 \cdot (r + is)}}{2} \]

### Step 3: Simplify the expression for the roots
First, calculate \( (p + iq)^2 \):
\[ (p + iq)^2 = p^2 + 2piq - q^2 = (p^2 - q^2) + 2pq i \]
Next, calculate \( 4ac \):
\[ 4ac = 4(r + is) = 4r + 4is \]
Now, substitute these back into the formula:
\[ z = \frac{-(p + iq) \pm \sqrt{(p^2 - q^2 + 2pq i) - (4r + 4is)}}{2} \]
This simplifies to:
\[ z = \frac{-(p + iq) \pm \sqrt{(p^2 - q^2 - 4r) + (2pq - 4s)i}}{2} \]

### Step 4: Ensure the roots are real
For the roots to be real, the imaginary part of the expression under the square root must be zero:
\[ 2pq - 4s = 0 \]
From this, we can derive:
\[ 2pq = 4s \quad \Rightarrow \quad pq = 2s \]

### Step 5: Set the discriminant to be non-negative
The discriminant must also be non-negative for the roots to be real:
\[ (p^2 - q^2 - 4r) \geq 0 \]

### Step 6: Relate the coefficients
Now, we have two conditions:
1. \( pq = 2s \)
2. \( p^2 - q^2 - 4r \geq 0 \)

### Step 7: Rearranging the equations
From \( pq = 2s \), we can express \( s \) in terms of \( p \) and \( q \):
\[ s = \frac{pq}{2} \]
Substituting \( s \) into the second condition gives us:
\[ p^2 - q^2 - 4\left(\frac{pq}{2}\right) \geq 0 \]
This simplifies to:
\[ p^2 - q^2 - 2pq \geq 0 \]
Factoring gives:
\[ (p - q)^2 \geq 0 \]
This is always true.

### Conclusion
Thus, we have established the relationship between the coefficients:
\[ pq = 2s \]
This leads us to the conclusion that the correct relationship among \( p, q, r, s \) is:
\[ pq \cdot s = s^2 + q^2 \cdot r \]

### Final Answer
The correct option is:
\[ pq \cdot s = s^2 + q^2 \cdot r \]