Let `f(x)=ax^(2)+bx+c`, `g(x)=ax^(2)+px+q`, where `a`, `b`, `c`, `q`, `p in R` and `b ne p`. If their discriminants are equal and `f(x)=g(x)` has a root `alpha`, then

Options

A. `alpha` will be `A.M.` of the roots of `f(x)=0`, `g(x)=0`
B. `alpha` will be `G.M.` of the roots of `f(x)=0`, `g(x)=0`
C. `alpha` will be `A.M.` of the roots of `f(x)=0` or `g(x)=0`
D. `alpha` will be `G.M.` of the roots of `f(x)=0` or `g(x)=0`

Explanation

To solve the problem, we need to analyze the functions \( f(x) = ax^2 + bx + c \) and \( g(x) = ax^2 + px + q \) under the given conditions.

### Step 1: Set the equations equal to each other
Since \( f(x) = g(x) \) has a root \( \alpha \), we can set the two functions equal to each other:
\[ ax^2 + bx + c = ax^2 + px + q \]
This simplifies to:
\[ bx + c = px + q \]
Rearranging gives:
\[ (b - p)x + (c - q) = 0 \]

### Step 2: Solve for \( x \)
Since \( \alpha \) is a root, we can substitute \( x = \alpha \):
\[ (b - p)\alpha + (c - q) = 0 \]
From this, we can express \( \alpha \) as:
\[ \alpha = \frac{q - c}{b - p} \]

### Step 3: Equate the discriminants
The discriminants of both quadratic equations must be equal:
\[ D_f = b^2 - 4ac \quad \text{and} \quad D_g = p^2 - 4aq \]
Setting them equal gives:
\[ b^2 - 4ac = p^2 - 4aq \]
Rearranging yields:
\[ b^2 - p^2 = 4a(c - q) \]

### Step 4: Analyze the roots
The roots of \( f(x) = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D_f}}{2a} \]
The roots of \( g(x) = 0 \) are:
\[ x = \frac{-p \pm \sqrt{D_g}}{2a} \]

### Step 5: Find the Arithmetic Mean (AM) and Geometric Mean (GM)
The roots of \( f(x) = 0 \) are \( r_1 \) and \( r_2 \), and the roots of \( g(x) = 0 \) are \( s_1 \) and \( s_2 \).

The Arithmetic Mean (AM) of the roots of \( f(x) \) is:
\[ AM_f = \frac{r_1 + r_2}{2} = -\frac{b}{2a} \]
The Arithmetic Mean (AM) of the roots of \( g(x) \) is:
\[ AM_g = \frac{s_1 + s_2}{2} = -\frac{p}{2a} \]

### Step 6: Determine the relationship with \( \alpha \)
Since \( \alpha \) is a root of both equations, we can conclude that:
\[ \alpha = \frac{q - c}{b - p} \]
This suggests that \( \alpha \) is related to the means of the roots of both quadratics.

### Conclusion
Given that \( b \neq p \), and the discriminants are equal, we can conclude that \( \alpha \) is the Arithmetic Mean of the roots of both equations.

Thus, the correct option is:
**Option 1: \( \alpha \) will be the AM of the roots of \( f(x) = 0 \) and \( g(x) = 0 \)**.