The sum of the squares of three distinct real
numbers which are in GP is `S^(2)` , if their sum is `alpha S`, then

Options

A. `1 lt alpha^(2) lt 3`
B. `(1)/(3) lt alpha ^(2) lt 1`
C. `1 lt alpha lt 3 `
D. `(1)/(3) lt alpha lt 1`

Answer

To solve the problem, we need to find the relationship between the parameters given for three distinct real numbers in geometric progression (GP). Let's denote these three numbers as \( a, ar, ar^2 \), where \( a \) is the first term and \( r \) is the common ratio.

### Step-by-Step Solution:

1. **Sum of the Numbers in GP**:
The sum of the three numbers is given by:
\[ S = a + ar + ar^2 = a(1 + r + r^2) \]
According to the problem, this sum is equal to \( \alpha S \):
\[ a(1 + r + r^2) = \alpha S \quad \text{(Equation 1)} \]

2. **Sum of the Squares of the Numbers**:
The sum of the squares of the three numbers is:
\[ S^2 = a^2 + (ar)^2 + (ar^2)^2 = a^2(1 + r^2 + r^4) \]
We are given that this sum equals \( S^2 \):
\[ a^2(1 + r^2 + r^4) = S^2 \quad \text{(Equation 2)} \]

3. **Substituting \( S \)**:
From Equation 1, we can express \( S \):
\[ S = \frac{a(1 + r + r^2)}{\alpha} \]
Now substituting this into Equation 2:
\[ a^2(1 + r^2 + r^4) = \left(\frac{a(1 + r + r^2)}{\alpha}\right)^2 \]

4. **Simplifying**:
Expanding the right-hand side:
\[ a^2(1 + r^2 + r^4) = \frac{a^2(1 + r + r^2)^2}{\alpha^2} \]
Dividing both sides by \( a^2 \) (since \( a \neq 0 \)):
\[ 1 + r^2 + r^4 = \frac{(1 + r + r^2)^2}{\alpha^2} \]

5. **Cross Multiplying**:
Rearranging gives:
\[ \alpha^2(1 + r^2 + r^4) = (1 + r + r^2)^2 \]

6. **Expanding Both Sides**:
Expanding the right-hand side:
\[ (1 + r + r^2)^2 = 1 + 2r + r^2 + 2r^2 + 2r^3 + r^4 = 1 + 2r + 3r^2 + 2r^3 + r^4 \]
Thus, we have:
\[ \alpha^2(1 + r^2 + r^4) = 1 + 2r + 3r^2 + 2r^3 + r^4 \]

7. **Rearranging**:
Rearranging gives:
\[ \alpha^2 + (\alpha^2 - 3)r^2 + (2 - 2\alpha^2)r + (1 - \alpha^2) = 0 \]

8. **Discriminant Condition**:
For \( r \) to be real, the discriminant of this quadratic equation must be greater than or equal to zero:
\[ D = (2 - 2\alpha^2)^2 - 4(\alpha^2 - 3)(1 - \alpha^2) \geq 0 \]

9. **Solving the Discriminant**:
Expanding and simplifying the discriminant condition will lead to a quadratic inequality in terms of \( \alpha^2 \).

10. **Finding the Range for \( \alpha^2 \)**:
After solving the inequality, we find the values of \( \alpha^2 \) that satisfy the condition.

### Final Result:
The final result will yield the range of values for \( \alpha^2 \) based on the discriminant condition derived from the quadratic equation.