question_answer

The heights of two trees are x and y, where\[x>y\]. The top of the trees are at a distance z apart. If s is the shortest distance between the trees, then \[{{s}^{2}}\] is equal to -

A) \[{{x}^{2}}+{{y}^{2}}-{{z}^{2}}-2xy\]

B) \[{{x}^{2}}+{{y}^{2}}-{{z}^{2}}\]

C) \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-2xy\]

D) \[{{z}^{2}}-{{x}^{2}}-{{y}^{2}}+2xy\]

Correct Answer: D

Solution :

Now, by applying Pythagoras theorem in \[\Delta \]ABE, \[A{{E}^{2}}=A{{E}^{2}}+B{{E}^{2}}~{{z}^{2}}={{\left( x-y \right)}^{2}}+{{s}^{2}}\] \[\Rightarrow {{z}^{2}}={{x}^{2}}+{{y}^{2}}-2xy+{{s}^{2}}\] \[~\Rightarrow {{\mathbf{S}}^{\mathbf{2}}}={{\mathbf{z}}^{\mathbf{2}}}-{{x}^{\mathbf{2}}}-{{\mathbf{y}}^{\mathbf{2}}}+\mathbf{2}x\mathbf{y}\]