The locus of point of the plane passing through \((\alpha, \beta, \gamma)\) and intersect the axis in \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and the plane which is parallel to such plane is (A) \((\mathrm{x} / \alpha)+(\mathrm{y} / \beta)+(\mathrm{z} / \gamma)=1\) (B) \((\alpha / x)+(\beta / y)+(\gamma / z)=1\) (C) \(x+y+z=1\) (D) \(\mathrm{x}+\mathrm{y}+\mathrm{z}=\alpha \beta \gamma\)

In 3D geometry, the plane equation is crucial for describing the position of a flat surface within a three-dimensional space. To define a plane, you typically need a point through which the plane passes and a normal vector that is perpendicular to the plane. However, when the plane intersects the axes at known points, there is a convenient linear form such as \[\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1\]where the constants \(a\), \(b\), and \(c\) represent the intercepts of the plane on the x, y, and z-axes respectively. This equation signifies that any point \((x, y, z)\) on the plane satisfies the relation when substituted back into the equation. To further understand the plane equation, it is essential to solve for these intercepts by using a known point on the plane, as was demonstrated in the provided exercise solution. Once these values are determined, they provide a complete description of the plane within the 3D coordinate system.

Coordinate geometry, also known as analytic geometry, brings together algebra and geometry by using an ordered pair of numbers, the coordinates, to describe the positions of points in space. In 3D, this system extends to three dimensions and points are defined by triplets of numbers \((x, y, z)\), representing their position along the x, y, and z-axes respectively.The principles of coordinate geometry are core to understanding the locus of points that form shapes like lines, planes, and other geometrical figures. In the context of determining the plane equation, coordinate geometry plays a pivotal role in establishing the relationships between the plane and points contained therein, as well as its interactions with the coordinate axes.
For instance, in the exercise provided, by knowing that the plane intersects the coordinate axes at specific points, and by utilizing the coordinates of a given point, you can derive the plane's equation using coordinate geometry techniques.

The 3D coordinate system is an extension of the 2D Cartesian system and is used to specify the positions of points in three-dimensional space. It is composed of three mutually perpendicular axes: the x-axis (horizontal), the y-axis (vertical), and the z-axis (depth). Each point in this space can be identified by an ordered triplet \((x, y, z)\), representing the distances from the origin to the projections of the point on each of the three axes. Understanding this coordinate system is vital for visualizing and solving problems related to 3D geometry, such as finding the locus of a point on a given plane. When dealing with problems like the textbook exercise, conceptualizing how the plane relates to the axes and visualizing the points of intersection help in deriving the correct formula for the plane equation and, ultimately, the locus of points that lie on that plane.