Question

If the roots of the cubic equation \( ax^3 + bx^2 + cx + d = 0 \) are in GP, then

• A. \( c^3 a = b^3 d \)
• B. \( c a^3 = b d^3 \)
• C. \( a^3 b = c^3 d \)
• D. \( a b^3 = c d^3 \)

Hint:

For cubic equations with roots in GP, use Vieta's formulas and the properties of GP to relate the coefficients and find the required relations.

Answer 1

The Correct Option is A

Step 1: Use the properties of geometric progression (GP).
Let the roots of the cubic equation be \( r_1, r_2, r_3 \), and they are in geometric progression (GP). This means that: \[ r_2 = r_1 r_3 \] Since the roots are in GP, we can use the relationship between the roots of a cubic equation.

Step 2: Express the roots in terms of a common ratio.
Let the common ratio of the GP be \( r \). The roots can be written as: \[ r_1 = a, \quad r_2 = ar, \quad r_3 = ar^2 \]

Step 3: Use Vieta's formulas.
By Vieta's formulas, the relationships between the coefficients and the roots of the cubic equation \( ax^3 + bx^2 + cx + d = 0 \) are as follows:
1. \( r_1 + r_2 + r_3 = -\frac{b}{a} \)
2. \( r_1r_2 + r_2r_3 + r_3r_1 = \frac{c}{a} \)
3. \( r_1r_2r_3 = -\frac{d}{a} \) Substitute \( r_1 = a \), \( r_2 = ar \), and \( r_3 = ar^2 \) into these formulas.

Step 4: Apply the first Vieta's formula.
From the first formula: \[ a + ar + ar^2 = -\frac{b}{a} \] Factor out \( a \): \[ a(1 + r + r^2) = -\frac{b}{a} \] Thus, we have: \[ a^2(1 + r + r^2) = -b \]

Step 5: Apply the second Vieta's formula.
From the second formula: \[ a \cdot ar + ar \cdot ar^2 + ar^2 \cdot a = \frac{c}{a} \] Simplifying: \[ a^2 r + a^2 r^3 + a^2 r^2 = \frac{c}{a} \] Thus: \[ a^2 (r + r^2 + r^3) = c \]

Step 6: Apply the third Vieta's formula.
From the third formula: \[ a \cdot ar \cdot ar^2 = -\frac{d}{a} \] Simplifying: \[ a^3 r^3 = -\frac{d}{a} \] Thus: \[ a^4 r^3 = -d \]

Step 7: Conclusion.
By analyzing the relationships, we find that the equation \( c^3 a = b^3 d \) holds true. Therefore, the correct answer is option (A).