JEE MAIN - Mathematics (2021 - 26th August Evening Shift - No. 13)
A hall has a square floor of dimension 10 m $$\times$$ 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is $${\cos ^{ - 1}}{1 \over 5}$$, then the height of the hall (in meters) is :
_26th_August_Evening_Shift_en_13_1.png)
5
2$$\sqrt {10} $$
5$$\sqrt {3} $$
5$$\sqrt {2} $$
Explanation
$$A(\widehat j)\,.\,B(10\widehat i)$$
$$H(h\widehat j + 10\widehat k)$$
$$G(10\widehat i + h\widehat j + 10\widehat k)$$
$$\overrightarrow {AG} = 10\widehat i + h\widehat j + 10\widehat k$$
$$\overrightarrow {BH} = - 10\widehat i + h\widehat j + 10\widehat k$$
$$\cos \theta = {{\overrightarrow {AG} \overrightarrow {BH} } \over {\left| {\overrightarrow {AG} } \right|\left| {\overrightarrow {BH} } \right|}}$$
$${1 \over 5} = {{{h^2}} \over {{h^2} + 200}}$$
$$4{h^2} = 200 \Rightarrow h = 5\sqrt 2 $$
$$H(h\widehat j + 10\widehat k)$$
$$G(10\widehat i + h\widehat j + 10\widehat k)$$
$$\overrightarrow {AG} = 10\widehat i + h\widehat j + 10\widehat k$$
$$\overrightarrow {BH} = - 10\widehat i + h\widehat j + 10\widehat k$$
$$\cos \theta = {{\overrightarrow {AG} \overrightarrow {BH} } \over {\left| {\overrightarrow {AG} } \right|\left| {\overrightarrow {BH} } \right|}}$$
$${1 \over 5} = {{{h^2}} \over {{h^2} + 200}}$$
$$4{h^2} = 200 \Rightarrow h = 5\sqrt 2 $$
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