If `x^(2)-ax-21=0 and x^(2)-3ax+35=0, a gt 0` have a common root, then a is equal to :

Options

A. 1
B. 2
C. 4
D. 5

Answer

To solve the problem, we need to find the value of \( a \) such that the equations \( x^2 - ax - 21 = 0 \) and \( x^2 - 3ax + 35 = 0 \) have a common root. Let's denote the common root as \( p \).

### Step-by-Step Solution:

1. **Substituting the Common Root**:
Since \( p \) is a root of both equations, we can substitute \( p \) into both equations.
- From the first equation:
\[ p^2 - ap - 21 = 0 \quad \text{(1)} \]
- From the second equation:
\[ p^2 - 3ap + 35 = 0 \quad \text{(2)} \]

2. **Equating the Two Equations**:
Since both equations equal zero, we can set them equal to each other:
\[ p^2 - ap - 21 = p^2 - 3ap + 35 \]

3. **Canceling \( p^2 \)**:
We can cancel \( p^2 \) from both sides:
\[ -ap - 21 = -3ap + 35 \]

4. **Rearranging the Equation**:
Rearranging gives:
\[ -ap + 3ap = 35 + 21 \]
\[ 2ap = 56 \]

5. **Solving for \( a \)**:
Dividing both sides by \( 2p \) (assuming \( p \neq 0 \)):
\[ a = \frac{56}{2p} = \frac{28}{p} \quad \text{(3)} \]

6. **Substituting \( a \) Back into One of the Equations**:
We can substitute \( a \) back into the first equation (1):
\[ p^2 - \left(\frac{28}{p}\right)p - 21 = 0 \]
Simplifying this gives:
\[ p^2 - 28 - 21 = 0 \]
\[ p^2 - 49 = 0 \]

7. **Finding the Value of \( p \)**:
This can be factored as:
\[ (p - 7)(p + 7) = 0 \]
Thus, \( p = 7 \) or \( p = -7 \). Since \( a > 0 \), we will use \( p = 7 \).

8. **Calculating \( a \)**:
Substituting \( p = 7 \) back into equation (3):
\[ a = \frac{28}{7} = 4 \]

### Final Answer:
Thus, the value of \( a \) is \( \boxed{4} \).