From the point (4 , 3) a perpendicular is dropped on the X-axis as well as on the Y-axis . If the lengths of perpendiculars are p and q respectively , then which one of the following is correct ?

Options

A. p = q
B. 3p = 4q
C. 4 p = 3q
D. p + q = 5

Answer

To solve the problem, we need to find the lengths of the perpendiculars dropped from the point (4, 3) to the X-axis and Y-axis.

### Step-by-Step Solution:

1. **Identify the point and axes**: The point given is (4, 3). The X-axis is the horizontal line where y = 0, and the Y-axis is the vertical line where x = 0.

2. **Calculate the length of the perpendicular to the X-axis (p)**:
- The perpendicular from the point (4, 3) to the X-axis will have the same x-coordinate but a y-coordinate of 0. Therefore, the coordinates of the foot of the perpendicular on the X-axis will be (4, 0).
- The length of this perpendicular (p) is the difference in the y-coordinates:
\[ p = 3 - 0 = 3 \]

3. **Calculate the length of the perpendicular to the Y-axis (q)**:
- The perpendicular from the point (4, 3) to the Y-axis will have the same y-coordinate but an x-coordinate of 0. Therefore, the coordinates of the foot of the perpendicular on the Y-axis will be (0, 3).
- The length of this perpendicular (q) is the difference in the x-coordinates:
\[ q = 4 - 0 = 4 \]

4. **Establish the relationship between p and q**:
- We have found that \( p = 3 \) and \( q = 4 \).
- To find a relationship, we can express these lengths in terms of a ratio or equation. For example:
\[ 4p = 12 \quad \text{(since } p = 3\text{)} \]
\[ 3q = 12 \quad \text{(since } q = 4\text{)} \]
- From these, we can see that:
\[ 4p = 3q \]

5. **Conclusion**:
- The correct relationship derived from the lengths of the perpendiculars is \( 4p = 3q \).

### Final Answer:
The correct option is \( 4p = 3q \).