Two vertical poles AL and BM of height 4 m and 16 m respectively stand apart on a horizontal plane. If A, B be the feet of the poles and AM and BL intersect at P, then the height of P from the horizontal plane is equal to

Options

A. 3.2 m
B. 2.5 m
C. 4 m
D. 8 m

Answer

To find the height of point P from the horizontal plane where lines AM and BL intersect, we can follow these steps:

### Step 1: Define the Points
Let:
- A be the foot of pole AL at coordinates (0, 0).
- L be the top of pole AL at coordinates (0, 4) since the height of pole AL is 4 m.
- B be the foot of pole BM at coordinates (L, 0) where L is the distance between points A and B.
- M be the top of pole BM at coordinates (L, 16) since the height of pole BM is 16 m.

### Step 2: Find the Equations of Lines AM and BL
1. **Equation of line AM** (from A to M):
- The slope of line AM = (y2 - y1) / (x2 - x1) = (16 - 0) / (L - 0) = 16/L.
- Using point-slope form: y - 0 = (16/L)(x - 0) → y = (16/L)x.

2. **Equation of line BL** (from B to L):
- The slope of line BL = (4 - 0) / (0 - L) = -4/L.
- Using point-slope form: y - 0 = (-4/L)(x - L) → y = -4/L(x - L) → y = -4/Lx + 4.

### Step 3: Set the Equations Equal to Find Intersection Point P
To find the coordinates of point P, we set the equations of lines AM and BL equal to each other:
\[ \frac{16}{L}x = -\frac{4}{L}x + 4. \]

### Step 4: Solve for x
Multiply through by L to eliminate the denominator:
\[ 16x = -4x + 4L. \]
Combine like terms:
\[ 16x + 4x = 4L \implies 20x = 4L \implies x = \frac{L}{5}. \]

### Step 5: Substitute x back to find y
Substituting \(x = \frac{L}{5}\) into the equation of line AM:
\[ y = \frac{16}{L}\left(\frac{L}{5}\right) = \frac{16}{5}. \]

### Step 6: Conclusion
The coordinates of point P are \(\left(\frac{L}{5}, \frac{16}{5}\right)\). The height of point P from the horizontal plane is the y-coordinate, which is:
\[ \frac{16}{5} \text{ m} = 3.2 \text{ m}. \]

### Final Answer
The height of point P from the horizontal plane is **3.2 m**.

---