If the roots of the equation `1/(x+p)+1/(x+q)=1/r` are equal in magnitude and opposite in sign, then
(A) p+q=r
(B)p+q=2r
(C) product of roots=-`1/2(p^2+q^2)`
(D) sum of roots=1

Options

A. `-(1)/(2)(p^(2)-q^(2))`
B. `p^(2)+q^(2)`
C. `(1)/(2)(p^(2)-q^(2))`
D. `-(1)/(2)(p^(2)+q^(2))`

Explanation

To solve the equation \( \frac{1}{x+p} + \frac{1}{x+q} = \frac{1}{r} \) under the condition that the roots are equal in magnitude and opposite in sign, we can follow these steps:

### Step 1: Combine the fractions on the left side
We start by taking the least common multiple (LCM) of the denominators:

\[ \frac{(x+q) + (x+p)}{(x+p)(x+q)} = \frac{1}{r} \]

This simplifies to:

\[ \frac{2x + p + q}{(x+p)(x+q)} = \frac{1}{r} \]

### Step 2: Cross multiply
Cross multiplying gives us:

\[ r(2x + p + q) = (x+p)(x+q) \]

### Step 3: Expand both sides
Expanding the right-hand side:

\[ r(2x + p + q) = x^2 + (p+q)x + pq \]

Expanding the left-hand side:

\[ 2rx + rp + rq = x^2 + (p+q)x + pq \]

### Step 4: Rearranging to form a quadratic equation
Rearranging the equation gives us:

\[ x^2 + (p + q - 2r)x + (pq - rp - rq) = 0 \]

### Step 5: Identify the sum and product of roots
For a quadratic equation of the form \( ax^2 + bx + c = 0 \):

- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

Here, \( a = 1 \), \( b = p + q - 2r \), and \( c = pq - rp - rq \).

### Step 6: Use the condition of roots
Since the roots are equal in magnitude and opposite in sign, we have:

\[ \alpha + \beta = 0 \]

This implies:

\[ p + q - 2r = 0 \implies p + q = 2r \]

### Step 7: Find the product of the roots
Now, we calculate the product of the roots:

\[ \alpha \beta = pq - rp - rq \]

Substituting \( r = \frac{p + q}{2} \):

\[ \alpha \beta = pq - \left(\frac{p + q}{2}\right)p - \left(\frac{p + q}{2}\right)q \]

This simplifies to:

\[ pq - \frac{p^2 + pq + q^2 + pq}{2} = pq - \frac{p^2 + 2pq + q^2}{2} \]

This can be rewritten as:

\[ pq - \frac{(p + q)^2}{2} = -\frac{1}{2}(p^2 + q^2) \]

### Conclusion
Thus, we have derived the following relationships:

1. \( p + q = 2r \) (Option B)
2. The product of the roots is \( -\frac{1}{2}(p^2 + q^2) \) (Option C)

### Final Answer
The correct options are:
- (B) \( p + q = 2r \)
- (C) Product of roots = \( -\frac{1}{2}(p^2 + q^2) \)