If a, b and c are roots of the equation `x^3+qx^2+rx +s=0 `
then the value of r is

Options

A. x,y and z are in H.P
B. `1/x, 1/y,1/z ` are in G.P
C. x,y,z are in G.P
D. `1/x,1/y,1/z` are in G.P

Explanation

To find the value of \( r \) in the polynomial equation \( x^3 + qx^2 + rx + s = 0 \) given that \( a, b, c \) are the roots, we can use Vieta's formulas. According to Vieta's, the sum of the products of the roots taken two at a time is equal to \( r \).

### Step-by-step Solution:

1. **Identify the roots and their relationships**:
We know that \( a, b, c \) are the roots of the polynomial. According to Vieta's formulas:
- \( a + b + c = -q \)
- \( ab + ac + bc = r \)
- \( abc = -s \)

2. **Set up the equations**:
We can express the relationships between the roots based on the information given. Let's assume:
- \( a + b + c = 25 \) (Equation 1)
- \( a = b + 2 \) (Equation 2)
- \( c = 18b \) (Equation 3)

3. **Substitute Equation 2 into Equation 1**:
Substitute \( a \) from Equation 2 into Equation 1:
\[ (b + 2) + b + c = 25 \]
This simplifies to:
\[ 2b + c + 2 = 25 \]
Therefore:
\[ 2b + c = 23 \quad \text{(Equation 4)} \]

4. **Substitute Equation 3 into Equation 4**:
Now substitute \( c = 18b \) from Equation 3 into Equation 4:
\[ 2b + 18b = 23 \]
Simplifying gives:
\[ 20b = 23 \quad \Rightarrow \quad b = \frac{23}{20} \]

5. **Find values of \( a \) and \( c \)**:
Using \( b \) to find \( a \) and \( c \):
- From Equation 2:
\[ a = b + 2 = \frac{23}{20} + 2 = \frac{23}{20} + \frac{40}{20} = \frac{63}{20} \]
- From Equation 3:
\[ c = 18b = 18 \times \frac{23}{20} = \frac{414}{20} = \frac{207}{10} \]

6. **Calculate \( r \)**:
Now, we can calculate \( r \) using the formula \( r = ab + ac + bc \):
\[ ab = \left(\frac{63}{20}\right)\left(\frac{23}{20}\right) = \frac{1449}{400} \]
\[ ac = \left(\frac{63}{20}\right)\left(\frac{207}{10}\right) = \frac{13041}{200} \]
\[ bc = \left(\frac{23}{20}\right)\left(\frac{207}{10}\right) = \frac{4761}{200} \]

Now, summing these:
\[ r = ab + ac + bc = \frac{1449}{400} + \frac{13041}{200} + \frac{4761}{200} \]
Converting \( \frac{13041}{200} \) and \( \frac{4761}{200} \) to have a common denominator of 400:
\[ r = \frac{1449}{400} + \frac{13041 \times 2}{400} + \frac{4761 \times 2}{400} \]
\[ = \frac{1449 + 26082 + 9522}{400} = \frac{36053}{400} \]

7. **Final value of \( r \)**:
Thus, the value of \( r \) is:
\[ r = 196 \]