The base `B C` of a ` A B C` is bisected at the point `(p ,q)` & the equation to the side `A B&A C` are `p x+q y=1` & `q x+p y=1` . The equation of the median through `A` is: `(p-2q)x+(q-2p)y+1=0` `(p+q)(x+y)-2=0` `(2p q-1)(p x+q y-1)=(p^2+q^2-1)(q x+p y-1)` none of these

Options

A. `qx - py = 0`
B. `(x)/(p) +(y)/(q) = 2`
C. `(2pq -1) (px +qy-1) =(p^(2)+q^(2)-1) (qx+py-1)`
D. `(p-2q)x+(q-2p)y = p^(2) +r^(2)`

Explanation

To solve the problem, we need to find the equation of the median through point A in triangle ABC, given that the base BC is bisected at point (p, q) and the equations of sides AB and AC are provided.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- The base BC is bisected at point (p, q).
- The equations of sides AB and AC are:
- \( px + qy = 1 \) (Equation of line AB)
- \( qx + py = 1 \) (Equation of line AC)

2. **Finding the Equation of the Median:**
- The median from vertex A to the midpoint of BC (which is (p, q)) can be derived using the concept of linear combinations of the equations of lines AB and AC.
- The general form for the equation of the median can be expressed as:
\[ L_1 + \lambda L_2 = 0 \]
where \( L_1 \) and \( L_2 \) are the equations of lines AB and AC respectively.

3. **Setting Up the Equation:**
- We can write:
\[ px + qy - 1 + \lambda (qx + py - 1) = 0 \]
- Rearranging gives:
\[ (p + \lambda q)x + (q + \lambda p)y - (1 + \lambda) = 0 \]

4. **Substituting the Midpoint (p, q):**
- Since the median passes through the point (p, q), we substitute \( x = p \) and \( y = q \) into the equation:
\[ (p + \lambda q)p + (q + \lambda p)q - (1 + \lambda) = 0 \]
- This simplifies to:
\[ p^2 + \lambda pq + q^2 + \lambda pq - (1 + \lambda) = 0 \]
- Combining like terms gives:
\[ p^2 + q^2 + 2\lambda pq - (1 + \lambda) = 0 \]

5. **Solving for λ:**
- Rearranging gives:
\[ 2\lambda pq - \lambda = 1 - (p^2 + q^2) \]
- Factoring out λ:
\[ \lambda(2pq - 1) = 1 - (p^2 + q^2) \]
- Solving for λ:
\[ \lambda = \frac{1 - (p^2 + q^2)}{2pq - 1} \]

6. **Substituting λ back into the Median Equation:**
- Now substitute λ back into the median equation:
\[ (p + \frac{1 - (p^2 + q^2)}{2pq - 1} q)x + (q + \frac{1 - (p^2 + q^2)}{2pq - 1} p)y - (1 + \frac{1 - (p^2 + q^2)}{2pq - 1}) = 0 \]
- After simplification, we arrive at the final equation of the median.

7. **Identifying the Correct Option:**
- After simplifying the median equation, we can compare it with the provided options:
- The correct option is:
\[ (2pq - 1)(px + qy - 1) = (p^2 + q^2 - 1)(qx + py - 1) \]

### Final Answer:
The equation of the median through A is:
\[ (2pq - 1)(px + qy - 1) = (p^2 + q^2 - 1)(qx + py - 1) \]