The condition that the roots of `x^3 -bx^2 +cx -d=0` are in G.P is

Options

A. `c^3 =b^3 d`
B. `c^2=b^2d`
C. `c=bd^3`
D. `c=bd^2`

Answer

To find the condition that the roots of the cubic equation \(x^3 - bx^2 + cx - d = 0\) are in geometric progression (G.P.), we can follow these steps:

### Step 1: Define the Roots
Let the roots of the equation be \(a\), \(\frac{a}{r}\), and \(ar\), where \(a\) is a non-zero constant and \(r\) is the common ratio of the G.P.

### Step 2: Use Vieta's Formulas
According to Vieta's formulas for a cubic equation \(x^3 + px^2 + qx + r = 0\):
- The sum of the roots is equal to the coefficient of \(x^2\) taken with the opposite sign.
- The sum of the products of the roots taken two at a time is equal to the coefficient of \(x\).
- The product of the roots is equal to the constant term taken with the opposite sign.

From our equation \(x^3 - bx^2 + cx - d = 0\):
1. The sum of the roots:
\[ a + \frac{a}{r} + ar = b \]
This can be rewritten as:
\[ a \left(1 + \frac{1}{r} + r\right) = b \quad \text{(Equation 1)} \]

### Step 3: Sum of the Products of Roots Taken Two at a Time
2. The sum of the products of the roots taken two at a time:
\[ a \cdot \frac{a}{r} + a \cdot ar + \frac{a}{r} \cdot ar = c \]
This simplifies to:
\[ \frac{a^2}{r} + a^2 + \frac{a^2 r}{1} = c \]
Factoring out \(a^2\), we get:
\[ a^2 \left(\frac{1}{r} + 1 + r\right) = c \quad \text{(Equation 2)} \]

### Step 4: Product of the Roots
3. The product of the roots:
\[ a \cdot \frac{a}{r} \cdot ar = d \]
This simplifies to:
\[ a^3 = d \quad \text{(Equation 3)} \]

### Step 5: Relate Equations
From Equation 1, we can express \(1 + \frac{1}{r} + r\) in terms of \(b\) and \(a\):
\[ 1 + \frac{1}{r} + r = \frac{b}{a} \]
From Equation 2, we can express \(1 + \frac{1}{r} + r\) in terms of \(c\) and \(a^2\):
\[ 1 + \frac{1}{r} + r = \frac{c}{a^2} \]

### Step 6: Equate the Two Expressions
Since both expressions equal \(1 + \frac{1}{r} + r\), we can set them equal to each other:
\[ \frac{b}{a} = \frac{c}{a^2} \]
Cross-multiplying gives:
\[ b a = c \]
Thus, we can express \(a\) as:
\[ a = \frac{c}{b} \quad \text{(Equation 4)} \]

### Step 7: Substitute Back into Product Equation
Now substitute Equation 4 into Equation 3:
\[ \left(\frac{c}{b}\right)^3 = d \]
This simplifies to:
\[ \frac{c^3}{b^3} = d \]
Rearranging gives us the final condition:
\[ c^3 = b^3 d \]

### Final Condition
The condition that the roots of the equation \(x^3 - bx^2 + cx - d = 0\) are in G.P. is:
\[ c^3 = b^3 d \]