A variable line through (p,q) cuts the axes of co ordinates at A and B respectively. Lines are drawn through A parallel to y-axis and through B parallel to x-axis. If they meet at P, then locus of p is

Options

A. `x/p+y/q=1`
B. `p/x+q/y=1`
C. `x/q+y/p=1`
D. `q/x+p/y=1`

Answer

To solve the problem, we need to find the locus of the point P, which is the intersection of the lines drawn through points A and B, where A and B are the points where a variable line through (p, q) intersects the x-axis and y-axis, respectively.

### Step-by-Step Solution:

1. **Identify Points A and B**:
- Let the variable line through point (p, q) intersect the x-axis at point A and the y-axis at point B.
- The coordinates of A will be (a, 0) and the coordinates of B will be (0, b).

2. **Equation of the Line**:
- The equation of the line that passes through point (p, q) and intersects the axes at A and B can be expressed in intercept form:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]

3. **Substituting Coordinates**:
- Since the line passes through the point (p, q), we can substitute these coordinates into the line equation:
\[ \frac{p}{a} + \frac{q}{b} = 1 \]

4. **Expressing a and b**:
- From the equation, we can express a and b in terms of p and q:
\[ a = \frac{p}{1 - \frac{q}{b}} \quad \text{and} \quad b = \frac{q}{1 - \frac{p}{a}} \]

5. **Finding the Coordinates of Point P**:
- The point P is defined as the intersection of the line drawn from A parallel to the y-axis and the line drawn from B parallel to the x-axis.
- The coordinates of point P will be (a, b).

6. **Finding the Locus**:
- To find the locus of point P, we need to eliminate the parameters a and b from the equation:
\[ \frac{p}{a} + \frac{q}{b} = 1 \]
- Rearranging gives:
\[ p = a(1 - \frac{q}{b}) \quad \text{and} \quad q = b(1 - \frac{p}{a}) \]

7. **Final Locus Equation**:
- By substituting \( a \) and \( b \) back into the equation, we get:
\[ \frac{p}{x} + \frac{q}{y} = 1 \]
- This represents the locus of point P.

### Conclusion:
The locus of the point P is given by the equation:
\[ \frac{p}{x} + \frac{q}{y} = 1 \]