If `alpha, beta` are the roots of `ax^(2)+bx+c=0 and alpha+k, beta+k` are the roots of `px^(2)+qx+r=0`, then `k=`

Options

A. `-(1)/(2)((a)/(b)-(p)/(q))`
B. `((a)/(b)-(p)/(q))`
C. `(1)/(2)((b)/(a)-(q)/(p))`
D. `(ab-pq)`

Answer

To find the value of \( k \) given the roots of two quadratic equations, we can follow these steps:

### Step 1: Identify the roots of the first equation
The first equation is given as:
\[ ax^2 + bx + c = 0 \]
The roots of this equation are \( \alpha \) and \( \beta \).

### Step 2: Use the relationship for the sum of the roots
From Vieta's formulas, the sum of the roots \( \alpha + \beta \) can be expressed as:
\[ \alpha + \beta = -\frac{b}{a} \]

### Step 3: Identify the roots of the second equation
The second equation is given as:
\[ px^2 + qx + r = 0 \]
The roots of this equation are \( \alpha + k \) and \( \beta + k \).

### Step 4: Use the relationship for the sum of the roots of the second equation
Again, using Vieta's formulas, the sum of the roots \( (\alpha + k) + (\beta + k) \) can be expressed as:
\[ (\alpha + k) + (\beta + k) = -\frac{q}{p} \]
This simplifies to:
\[ \alpha + \beta + 2k = -\frac{q}{p} \]

### Step 5: Substitute the sum of the roots from the first equation
Now we can substitute the expression for \( \alpha + \beta \) from Step 2 into the equation from Step 4:
\[ -\frac{b}{a} + 2k = -\frac{q}{p} \]

### Step 6: Solve for \( k \)
Rearranging the equation to isolate \( k \):
\[ 2k = -\frac{q}{p} + \frac{b}{a} \]
\[ k = \frac{-\frac{q}{p} + \frac{b}{a}}{2} \]
This can be simplified further:
\[ k = \frac{b}{2a} - \frac{q}{2p} \]

### Final Answer
Thus, the value of \( k \) is:
\[ k = \frac{b}{2a} - \frac{q}{2p} \]