If a and c are such that the quadratic equation `ax^(2)-5x+c=0` has 10 as the sum of the roots and also as the product of the roots, find a and c respectively :

Options

A. `(1)/(2), 6`
B. `(1)/(2), 5`
C. `(1)/(2), 8`
D. none of these

Answer

To solve the problem, we need to find the values of \( a \) and \( c \) in the quadratic equation \( ax^2 - 5x + c = 0 \) given that the sum and product of the roots are both equal to 10.

### Step-by-Step Solution:

1. **Identify the Relationship Between Coefficients and Roots**:
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} \]
and the product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} \]

2. **Substituting the Given Values**:
In our equation, \( b = -5 \), so:
\[ \alpha + \beta = -\frac{-5}{a} = \frac{5}{a} \]
We are given that the sum of the roots is 10:
\[ \frac{5}{a} = 10 \]

3. **Solving for \( a \)**:
To find \( a \), we can cross-multiply:
\[ 5 = 10a \]
Dividing both sides by 10 gives:
\[ a = \frac{5}{10} = \frac{1}{2} \]

4. **Finding \( c \)**:
Now, we use the product of the roots:
\[ \alpha \beta = \frac{c}{a} \]
We know that the product of the roots is also 10:
\[ \frac{c}{a} = 10 \]
Substituting \( a = \frac{1}{2} \):
\[ \frac{c}{\frac{1}{2}} = 10 \]
This can be rewritten as:
\[ c \cdot 2 = 10 \]
Thus, dividing both sides by 2 gives:
\[ c = 5 \]

5. **Final Values**:
We have found:
\[ a = \frac{1}{2}, \quad c = 5 \]

### Conclusion:
The values of \( a \) and \( c \) are:
\[ \boxed{\left(\frac{1}{2}, 5\right)} \]