What are the points on the axis of x whose perpendicular distance from the straight line x/p + y/q = 1 is p?

Options

A. `(p)/(q)[q+sqrt((p^(2)+q^(2))),0`
B. `(p)/(q)[q-sqrt((p^(2)+q^(2))),0`
C. Both (a) and (b)
D. None of these

Answer

To solve the problem, we need to find the points on the x-axis whose perpendicular distance from the line given by the equation \( \frac{x}{p} + \frac{y}{q} = 1 \) is equal to \( p \).

### Step-by-step Solution:

1. **Rewrite the Line Equation**:
The equation of the line can be rewritten in standard form:
\[ \frac{x}{p} + \frac{y}{q} = 1 \implies x \cdot q + y \cdot p = pq \]
This gives us the line in the form \( Ax + By + C = 0 \) where \( A = q \), \( B = p \), and \( C = -pq \).

2. **Identify the Point on the X-axis**:
A point on the x-axis can be represented as \( (t, 0) \), where \( t \) is the x-coordinate we need to find.

3. **Calculate the Perpendicular Distance**:
The formula for the perpendicular distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \( (t, 0) \) into the formula:
\[ d = \frac{|qt + p \cdot 0 - pq|}{\sqrt{q^2 + p^2}} = \frac{|qt - pq|}{\sqrt{q^2 + p^2}} \]

4. **Set the Distance Equal to p**:
According to the problem, this distance is equal to \( p \):
\[ \frac{|qt - pq|}{\sqrt{q^2 + p^2}} = p \]

5. **Remove the Modulus**:
We can remove the modulus by considering two cases:
\[ qt - pq = p\sqrt{q^2 + p^2} \quad \text{(1)} \]
\[ qt - pq = -p\sqrt{q^2 + p^2} \quad \text{(2)} \]

6. **Solve for t in Both Cases**:
- For equation (1):
\[ qt = pq + p\sqrt{q^2 + p^2} \]
\[ t = \frac{pq + p\sqrt{q^2 + p^2}}{q} \]

- For equation (2):
\[ qt = pq - p\sqrt{q^2 + p^2} \]
\[ t = \frac{pq - p\sqrt{q^2 + p^2}}{q} \]

7. **Combine the Results**:
Thus, the points on the x-axis are:
\[ t = \frac{p}{q} \left( q \pm \sqrt{q^2 + p^2} \right) \]

### Final Answer:
The points on the x-axis whose perpendicular distance from the line \( \frac{x}{p} + \frac{y}{q} = 1 \) is \( p \) are:
\[ \left( \frac{p}{q} \left( q + \sqrt{p^2 + q^2} \right), 0 \right) \quad \text{and} \quad \left( \frac{p}{q} \left( q - \sqrt{p^2 + q^2} \right), 0 \right) \]