If m and n are roots of the equation `(x +p) (x + q) - k = 0`. Then roots of the equation `(x - m) (x -n) + k = 0` are

Options

A. a) p and q
B. b) `(1)/(p) and (1)/(q)`
C. c) `-p and -q`
D. d) `p + and p - q`

Answer

To solve the problem, we need to analyze the given equations step by step.

### Step 1: Understand the first equation
The first equation is given as:
\[ (x + p)(x + q) - k = 0 \]
This can be expanded to:
\[ x^2 + (p + q)x + (pq - k) = 0 \]
Here, the coefficients are:
- \(a = 1\)
- \(b = p + q\)
- \(c = pq - k\)

### Step 2: Identify the roots
The roots \(m\) and \(n\) of the equation can be found using Vieta's formulas:
- The sum of the roots \(m + n = -\frac{b}{a} = -(p + q)\)
- The product of the roots \(mn = \frac{c}{a} = pq - k\)

### Step 3: Write the second equation
The second equation is:
\[ (x - m)(x - n) + k = 0 \]
Expanding this gives:
\[ x^2 - (m + n)x + mn + k = 0 \]

### Step 4: Substitute the values of \(m + n\) and \(mn\)
From Step 2, we substitute:
- \(m + n = -(p + q)\)
- \(mn = pq - k\)

Thus, the equation becomes:
\[ x^2 - (-(p + q))x + (pq - k + k) = 0 \]
This simplifies to:
\[ x^2 + (p + q)x + pq = 0 \]

### Step 5: Identify the roots of the new equation
Now we can find the roots of the new equation \(x^2 + (p + q)x + pq = 0\) using Vieta's formulas again:
- The sum of the roots is \(-(p + q)\)
- The product of the roots is \(pq\)

The roots of this equation are \( -p \) and \( -q \).

### Final Answer
Thus, the roots of the equation \((x - m)(x - n) + k = 0\) are:
\[ \text{Roots are } -p \text{ and } -q. \]

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