Two vertical poles of height a and b subtend the same angle `45^(@)` at a point on the line joining their feet, the square of the distance between their tops is

Options

A. `(1//2) (a^(2) + b^(2))`
B. `a^(2) + b^(2)`
C. `2 (a^(2) + b^(2))`
D. `(a + b)^(2)`

Answer

To solve the problem of finding the square of the distance between the tops of two vertical poles of heights \( a \) and \( b \) that subtend the same angle of \( 45^\circ \) at a point on the line joining their feet, we can follow these steps:

### Step 1: Understand the Geometry
We have two vertical poles:
- Pole 1 with height \( a \)
- Pole 2 with height \( b \)

Let the feet of the poles be points \( D \) and \( C \), respectively. The point \( E \) on the ground is such that both poles subtend an angle of \( 45^\circ \) at \( E \).

### Step 2: Set Up the Triangles
From point \( E \):
- For pole 1 (height \( a \)):
\[ \tan(45^\circ) = \frac{a}{DE} \implies DE = a \]

- For pole 2 (height \( b \)):
\[ \tan(45^\circ) = \frac{b}{CE} \implies CE = b \]

### Step 3: Find the Distance Between the Feet of the Poles
The total distance \( DC \) between the feet of the poles is:
\[ DC = DE + CE = a + b \]

### Step 4: Calculate the Distance Between the Tops of the Poles
Let \( AE \) be the distance between the tops of the poles. We can apply the Pythagorean theorem in triangle \( ABE \):
\[ AE^2 = AB^2 + BE^2 \]
Where:
- \( AB = |b - a| \) (the vertical distance between the tops of the poles)
- \( BE = a + b \) (the horizontal distance between the feet of the poles)

### Step 5: Substitute the Values
Substituting the values we have:
\[ AE^2 = (b - a)^2 + (a + b)^2 \]

### Step 6: Expand the Equation
Now, we expand both squares:
\[ AE^2 = (b^2 - 2ab + a^2) + (a^2 + 2ab + b^2) \]
Combining like terms:
\[ AE^2 = b^2 - 2ab + a^2 + a^2 + 2ab + b^2 \]
\[ AE^2 = 2a^2 + 2b^2 \]

### Final Result
Thus, the square of the distance between the tops of the two poles is:
\[ AE^2 = 2(a^2 + b^2) \]