A square of side 'a' lies above the \(\mathrm{x}\) -axis and has one vertex at the origin. The side passing through the origin makes an angle o. \([0<\alpha<(\pi / 4)]\) with the positive direction of \(\mathrm{x}\) -axis. The equation of the diagonal not passing through the origin is (a) \(y(\cos \alpha-\sin \alpha)-x(\sin \alpha-\cos \alpha)=a\) (b) \(y(\cos \alpha+\sin \alpha)+x(\sin \alpha-\cos \alpha)=a\) (c) \(y(\cos \alpha-\sin \alpha)-x(\sin \alpha+\cos \alpha)=a\) (d) \(y(\cos \alpha+\sin \alpha)+x(\cos \alpha-\sin \alpha)=a\)

Understanding the equation of a diagonal in the context of coordinate geometry involves visualizing shapes and their properties within the coordinate system. In our specific problem involving a square, drawing the shape over the x-axis clarifies the positions of its vertices.

When dealing with coordinate geometry, always remember that any line segment or diagonal can be portrayed by an equation. Here, the initial step is to find the vertices of the square, which are determined by its side length 'a' and the angle \(\alpha\) it makes with the x-axis. Once the coordinates are known, the line connecting two vertices becomes a set of solutions to a linear equation.

It becomes crucial to understand how each vertex relates to one another and how, with a square's properties, you can deduce the position of the other vertices given one vertex. This process exemplifies the practical application of coordinate geometry in solving real-world geometric problems. Through coordinate geometry, equations are derived that represent line segments, which in this case, is the diagonal of a square.

Squares possess specific qualities that can simplify complex geometric problems. Each side of a square has the same length, and the angles between sides are always right angles (\(90^\circ\)). Moreover, the diagonals in a square are equal in length and bisect each other at right angles.

For the given exercise, where a square is oriented in a unique position in the coordinate plane, one must apply these square properties to find the coordinates of its vertices after a rotation. Recognizing that adjacent sides form a \(90^\circ\) angle, we can use trigonometric ratios to find the rotated vertex positions. Additionally, knowing a square's diagonal bisects the angles at the vertices helps to visualize the diagonal's equation in the coordinate plane.

Understanding these properties ensures that the correct method is applied when solving the equation of the diagonal. A solid grasp on the inherent characteristics of a square is invaluable, not just for this problem, but for any problem involving this fundamental shape.

Trigonometric ratios like sine and cosine are fundamental tools used to describe the relationship between the angles and sides of right-angled triangles. These ratios are also essential in the rotation of points on the coordinate plane.

In the context of this exercise, the side of the square at the origin forms a right angle triangle with the axes. Using trigonometric ratios, you can determine how far along the x and y axes the side of the square extends, given its length 'a' and the angle \(\alpha\). The cosine ratio gives us the horizontal component (x-coordinate) while the sine ratio gives the vertical component (y-coordinate).

Trigonometric ratios become immensely useful when we perform rotations. To find the third vertex, we rotate the second vertex by \(90^\circ\), which involves swapping the respective sine and cosine values and negating one to simulate a quarter-turn. This technique reflects a deep connection between trigonometry and coordinate geometry, showing how they can be used in tandem to solve problems related to geometry in a plane.