Consider three points \(P=(-\sin (\beta-\alpha),-\cos \beta)\) \(\mathrm{Q}=(\cos (\beta-\alpha), \sin \beta)\) and \(\mathrm{R}=(\cos (\beta-\alpha+\theta), \sin (\beta-0))\) where \(0<\alpha, \beta, \theta<(\pi / 4)\) then (a) P lies on the \(\underline{\mathrm{RQ}}\) (b) \(Q\) lie on the \(\underline{P R}\) (c) \(\mathrm{R}\) lie on the \(\mathrm{QP}\) (d) \(\mathrm{P}, \mathrm{Q}, \mathrm{R}\) are non collinear

Collinear points refer to a set of three or more points that lie on a single straight line. It is interesting to note that any two points are always collinear since a straight line can be drawn through any pair of points. However, the term is more meaningful when discussing three or more points.

To determine if additional points are collinear with the first two, specific conditions or formulas can be applied. In the context of coordinate geometry, one of the methods to check for collinearity is calculating the area of a triangle formed by the points. If the area is zero, this implies that the altitude is also zero and hence the points are on the same line – they are collinear. Otherwise, if the area is nonzero, this indicates that the points form a triangle of positive area, proving that they are not collinear.

In practical problems, this is often achieved by plugging the points' coordinates into the area formula and simplifying it. If the result is zero, the conclusion is that the points are collinear.

The area of a triangle in coordinate geometry can be computed using the determinantal formula involving the coordinates of its vertices. Given three points – A, B, and C – with coordinates \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\), the area (\(A_{triangle}\)) is calculated as:

\[A_{triangle} = \frac{1}{2}|(x_2 - x_1)(y_3 - y_1) - (x_3 - x_1)(y_2 - y_1)|.\]
By evaluating this formula, one can easily determine the area and utilize it for various purposes such as checking the collinearity of the points. A positive area indicates a definable shape, while an area of zero signifies that the points do not form anything other than a straight line.

The area formula is an essential tool not just for determining shapes, but also in broader applications across geometry and trigonometry, including finding the height, base lengths, and even utilizing these concepts in real-world situations such as land mapping.

Trigonometric coordinates refer to the representation of points on the Cartesian plane using trigonometric functions such as sine (sin), cosine (cos), or tangent (tan). These are particularly useful in scenarios where geometry and trigonometry intersect – such as defining points on a circle or other trigonometric paths.

When given an angle, say \(\theta\), points on a unit circle can be expressed in terms of \(\cos(\theta)\) for the x-coordinate and \(\sin(\theta)\) for the y-coordinate. The beauty of trigonometric coordinates lies in their ability to link circular or periodic properties to the familiar Cartesian plane, granting an elegant way to analyze circular motion or patterns through a linear perspective.

Understanding how to work with these coordinates is critical in not just geometry, but extends to fields like physics (think circular motion), engineering (signal processing, for example), and even in art (such as spirograph designs). It allows students to visualize and solve problems involving angles and lengths within various shapes and motions.