The Locus of the Point for which the Sum of Distances from the Coordinate Axes is 9

Consider a point in a two-dimensional Cartesian plane, where its positions on the x-axis and y-axis are denoted by x and y respectively. The fascinating mathematical exploration lies in the locus of the point (x, y) where the sum of the distances from the coordinate axes is exactly 9. To delve into this intriguing problem, we need to explore the geometric locus of such a point.

The Concept of Distance and Locus

In geometry, the concept of a locus refers to the collection of all points that satisfy given conditions. The distance from a point (x, y) to the x-axis is simply the absolute value of the y-coordinate, |y|. Similarly, the distance to the y-axis is |x|. The problem at hand revolves around finding these points (x, y) such that |x| |y| 9. Understanding this is crucial to grasp the geometric implications.

Graphing the Locus

The algebraic equation |x| |y| 9 dictates the locus of the point. To visualize this, we split the equation into different quadrants to consider positive and negative values of x and y.

First Quadrant (x 0, y 0)

In this quadrant, the equation simplifies to x y 9. This represents a line with a slope of -1 and an intercept of 9 on both the x-axis and the y-axis.

Second Quadrant (x 0, y 0)

In this quadrant, the equation becomes -x y 9, which is a line with a slope of 1 and an intercept of 9 on the y-axis. It intersects the x-axis at -9.

Third Quadrant (x 0, y 0)

In the third quadrant, the equation transforms into -x - y 9, representing a line with a slope of -1 and an intercept of -9 on both axes.

Fourth Quadrant (x 0, y 0)

Here, the equation simplifies to x - y 9, which is a line with a slope of 1 and an intercept of 9 on the x-axis. It crosses the y-axis at -9.

Visualizing the Locus

When these equations are graphed, the locus of the point (x, y) forms a square with vertices at (-9, 0), (0, 9), (9, 0), and (0, -9). The square is centered at the origin and extends symmetrically along the x-axis and y-axis.

Mathematical Significance and Applications

The concept of a locus with geometric constraints, such as the sum of distances to the coordinate axes being 9, finds applications in various fields. From architectural design to computer graphics, the understanding of these loci can significantly enhance problem-solving skills and provide deep insights into the interplay between algebra and geometry.

Architectural Design

In architectural design, understanding such loci can help in creating symmetrical structures and layouts. For instance, ensuring that certain sections of a building maintain a specific distance from the axes can lead to aesthetically pleasing and functional designs.

Computer Graphics and Animation

In the realm of computer graphics and animation, these loci can be used to create specific paths that objects must follow based on geometrical constraints. This can be particularly useful in creating animations where objects need to move in a predefined pattern.

Conclusion

The locus of the point for which the sum of the distances from the coordinate axes is 9 is a fascinating and instructive problem in geometry. It involves the application of absolute value and linear equations to derive a geometric figure that is both mathematically and visually significant. Understanding such concepts not only enriches one's problem-solving skills but also offers practical applications in fields like architecture and computer graphics.

Through this exploration, we gain a deeper appreciation for the elegance of mathematics and its wide-ranging applications. Whether it's in designing buildings or creating complex animations, the principles of geometric loci provide a strong foundation for creative and technological advancements.