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Step-by-Step Solution
Step 1: Identify the square and its vertices
Let the square have vertices A, B, C, and D, with vertex
A at the origin $(0,0)$. The side from the origin along the angle
$0 < \alpha < \frac{\pi}{4}$ has length $a$. Hence,
B has coordinates:
$$
B \bigl(a\cos\alpha,\;a\sin\alpha\bigr).
$$
Step 2: Determine the remaining vertices
The direction perpendicular to the vector $\bigl(\cos\alpha,\;\sin\alpha\bigr)$
is given by $\bigl(-\sin\alpha,\;\cos\alpha\bigr)$. Therefore, from the origin,
the vertex D is:
$$
D \bigl(-\,a\sin\alpha,\;a\cos\alpha\bigr).
$$
The remaining vertex C (the one diagonally opposite the origin) is the
vector sum of B and D:
$$
C = B + D = \bigl(a\cos\alpha,\;a\sin\alpha\bigr)
+ \bigl(-\,a\sin\alpha,\;a\cos\alpha\bigr)
= \bigl(a(\cos\alpha - \sin\alpha),\;a(\sin\alpha + \cos\alpha)\bigr).
$$
Step 3: Identify the diagonal not passing through the origin
The diagonal passing through the origin is $\overline{AC}$.
The diagonal not passing through the origin is $\overline{BD}$.
Hence, we must find the equation of the line through
$B(a\cos\alpha,\;a\sin\alpha)$ and $D(-\,a\sin\alpha,\;a\cos\alpha)$.
Step 4: Find the equation of line BD
One way to form the equation of the line through two points
$(x_{1},\,y_{1})$ and $(x_{2},\,y_{2})$ is to use the two-point form:
$$
(y - y_{1})(x_{2} - x_{1}) \;-\; (x - x_{1})(y_{2} - y_{1}) \;=\; 0.
$$
Here, let
$x_{1} = a\cos\alpha,\;y_{1} = a\sin\alpha$
and
$x_{2} = -\,a\sin\alpha,\;y_{2} = a\cos\alpha.$
Substituting:
\[
(y - a\sin\alpha)\bigl(-\,a\sin\alpha - a\cos\alpha\bigr)
\;-\; (x - a\cos\alpha)\bigl(a\cos\alpha - a\sin\alpha\bigr)
\;=\; 0.
\]
Factor out $a$ and simplify. After rearranging terms, it becomes:
$$
( \cos\alpha - \sin\alpha )\,x
\;+\; ( \sin\alpha + \cos\alpha )\,y
\;=\; a.
$$
Step 5: Compare with the given options
The equation we found matches the correct answer provided:
$$
y\,(\cos\alpha + \sin\alpha)
+ x\,(\cos\alpha - \sin\alpha)
= a.
$$
Conclusion
Therefore, the diagonal of the square not passing through the origin is given by
the equation
$$
y(\cos\alpha + \sin\alpha)
+ x(\cos\alpha - \sin\alpha)
= a,
$$
which is the correct choice.
