2021 NJC P1 Q5
Two straight corridors, $P$ and $Q$, each of width $2$ m, meet at right angles. A banner is hung across the ceiling of the corridors using a taut string such that the string is parallel to the ground and always touches the inside corner of the wall at point $B$. The string also touches the outer walls at variable points $A$ and $C$ respectively. In the position shown in the diagram, the acute angle between $AC$ and the wall of corridor $P$ is $\frac{\pi }{4}+\theta $, where $\theta $ is a sufficiently small angle.
(i)
Show that $AC=2\left[ \frac{1}{\sin \left( \frac{\pi }{4}+\theta \right)}+\frac{1}{\cos \left( \frac{\pi }{4}+\theta \right)} \right]$.
[2]
(ii)
Hence show that
$AC\approx r+s{{\theta }^{2}}$
where $r$ and $s$ are constants to be determined.
[5]
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