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A square of side a is placed in the upper half-plane with one corner at the origin. The side through the origin forms an angle α, where 0 < α < π/4, with the positive x-axis. Find the equation of the diagonal that does not pass through the origin.

1. y(cos α + sin α) + x(cos α − sin α) = a
2. y(cos α − sin α) − x(sin α − cos α) = a
3. y(cos α + sin α) + x(sin α − cos α) = a
4. y(cos α + sin α) + x(sin α + cos α) = a

Correct answer: y(cos α + sin α) + x(cos α − sin α) = a

Solution

The correct option represents the equation of the diagonal by combining the contributions of both the x and y coordinates, adjusted for the angle α. The terms reflect the orientation of the square in the upper half-plane, ensuring that the diagonal correctly accounts for the angle formed with the axes.

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