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Let $u$ be a point on the unit circle in the first quadrant (i.e., both coordinates of $u$ are positive). Let $\theta$ be the angle subtended by $u$ and the $x$ axis at the origin. Let $\ell _{u}$ denote the infinite line passing through the origin and $u$. Consider the following operation $O_{u}$ on points in the plane.

$\textbf{Operation }O_{u}$

$\textbf{INPUT:}$ a point $v$ on the plane

  1. Reflect $v$ in the $x$ axis, obtaining $\tilde{v}$.
  2. Reflect $\tilde{v}$ in $\ell_{u}$, obtaining $\hat{v}$.
  3. Output $\hat{v}$.

If $\hat{v}$ is the output of applying $O_{u}$ on $v$, we write $O_{u}(v) = \hat{v}$. Further, we denote by $O_{u}^{k}$ the iterates of $O_{u}$, i.e., $O_{u}^{1}(v):=O_{u}(v)$ and $O_{u}^{k}(v):=O_{u}(O_{u}^{k-1}(v))$ for all integers $k>1$.

Consider a point $v$ in the first quadrant such that $v$ and the $x$-axis subtend an angle $\phi$ at the origin. Define $w = O_{u}^{8}(v).$ Assuming $\theta = 5^{\circ}$ and $\phi = 10^{\circ}$, what is the angle subtended by $w$ and the $x$-axis at the origin?

  1. $50^{\circ}$
  2. $85^{\circ}$
  3. $90^{\circ}$
  4. $145^{\circ}$
  5. $165^{\circ}$
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The reflection matrix, to reflect a point $(x,y)$ about the line $y=mx$ where $m=tan\theta$ is given by,

$\begin{bmatrix} cos2\theta & sin2\theta\\ sin2\theta&-cos2\theta \end{bmatrix}$

So, if we have a point $(x,y)$ and after reflection, new point is $(x',y')$ then we can write it as :

$\begin{bmatrix} x'\\y' \end{bmatrix}$ =  $\begin{bmatrix} cos2\theta & sin2\theta\\ sin2\theta&-cos2\theta \end{bmatrix}$$\begin{bmatrix} x\\y \end{bmatrix}$

Now, here in the given question, given point is $(vcos\phi,vsin\phi)$

After reflecting about $x-$axis, It becomes, $(vcos\phi,-vsin\phi)$

Now, according to given information for $\theta = 5^{\circ}$ and $\phi = 10^{\circ}$,

$O_{u}(v)= v\begin{bmatrix} cos10 &sin10 \\ sin10 &-cos10 \end{bmatrix}\begin{bmatrix} cos10\\-sin10 \end{bmatrix}= v\begin{bmatrix} cos20\\sin20 \end{bmatrix}$

$O_{u}^{2}(v)= v\begin{bmatrix} cos10 &sin10 \\ sin10 &-cos10 \end{bmatrix}\begin{bmatrix} cos20\\-sin20 \end{bmatrix}= v\begin{bmatrix} cos30\\sin30 \end{bmatrix}$

$O_{u}^{3}(v)= v\begin{bmatrix} cos10 &sin10 \\ sin10 &-cos10 \end{bmatrix}\begin{bmatrix} cos30\\-sin30 \end{bmatrix}= v\begin{bmatrix} cos40\\sin40 \end{bmatrix}$

Similarly,

$O_{u}^{8}(v)= v\begin{bmatrix} cos10 &sin10 \\ sin10 &-cos10 \end{bmatrix}\begin{bmatrix} cos80\\-sin80 \end{bmatrix}= v\begin{bmatrix} cos90\\sin90 \end{bmatrix}$

It means new point is $w= (vcos90^{\circ}, vsin90^{\circ})$.

So, angle subtended by $w$ and $x-$axis at origin is $90^{\circ}.$
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