Multiple choice

$P$ is a point on the segment joining the feet of two vertical poles of heights $a$ and $b$. The angles of elevation of the tops of the poles from $P$ are $45^0$ each. Then, the square of the distance between the tops of the poles is

  1. $\displaystyle \frac {a^2\, +\, b^2}{2}$
  2. $a^2\, +\, b^2$
  3. $2(a^2\, +\, b^2)$
  4. $4(a^2\, +\, b^2)$
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C Correct answer
Explanation

Let the poles be at (0,0) and (d,0) with heights a and b. P is at (x,0). tan(45) = a/x = 1, so x=a. tan(45) = b/(d-x) = 1, so d-x=b, d=a+b. The tops are at (0,a) and (a+b,b). Distance squared = (a+b-0)^2 + (b-a)^2 = (a+b)^2 + (b-a)^2 = a^2 + 2ab + b^2 + b^2 - 2ab + a^2 = 2(a^2 + b^2).

AI explanation

Let the pole heights be a and b, and let P be at distances x and y from their respective bases. Since the angle of elevation to both tops is 45 degrees, tan 45 equals 1, meaning x equals a and y equals b. The poles are vertical and on opposite sides of P, so the horizontal distance between their tops is a plus b. Using the Pythagorean theorem, the square of the distance between the tops is (a plus b) squared plus (a minus b) squared. Expanding both squares gives a squared plus 2ab plus b squared plus a squared minus 2ab plus b squared, which simplifies to 2(a squared plus b squared).

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