Question

$P$ 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 ${45}^o$ each. Then, the square of the distance between the tops of the poles is

• A. $\frac{(a^2+b^2)}{2}$
• B. $\left(a^2+b^2\right)$
• C. $2(a^2+b^2)$
• D. $4(a^2+b^2)$

Answer 1

The correct option is C

$2(a^2+b^2)$

Explanation for the correct option:

Finding the value of $D{C}^2$:

We form a figure using the information given

Now according to the figure, we know the distance

$$\begin{gathered} OA=A \\ OB=B \\ DE=a+b \\ CE=b-a \end{gathered}$$

Now taking triangle $DEC$ we use Pythagoras theorem

$$\begin{gathered} D{C}^2=D{E}^2+C{E}^2 \\ D{C}^2={\left(a+b\right)}^2+{\left(b-a\right)}^2 \\ D{C}^2=a^2+b^2+2ab+b^2+a^2-2ab \\ D{C}^2=2\left(a^2+b^2\right) \end{gathered}$$

Therefore, the correct answer is option (C).