Problem 1 - Technical Mathematics with Calculus

Question

If \(h\) and \(k\) are positive quantities, in which quadrants would the following points lie? $$(h,-k)$$

Step-by-Step Solution

Answer

The point (h, -k), with h and k being positive, lies in Quadrant IV.

Step 1: Understand the Coordinate System

The coordinate system is divided into four quadrants. The signs of a point's coordinates determine its quadrant. Quadrant I has both coordinates positive, Quadrant II has negative x-coordinate and positive y-coordinate, Quadrant III has both coordinates negative, and Quadrant IV has a positive x-coordinate and negative y-coordinate.

Step 2: Analyze the Given Point

The given point is (h, -k) where h and k are positive quantities. This indicates that the x-coordinate 'h' is positive and the y-coordinate '-k' is negative.

Step 3: Determine the Quadrant of the Point

Since the x-coordinate is positive and the y-coordinate is negative, the point (h, -k) falls in Quadrant IV.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves studying geometry using a coordinate system. This system allows us to precisely locate points on a plane using a pair of numerical coordinates.
In a typical two-dimensional Cartesian coordinate system, there are two axes: the horizontal x-axis and the vertical y-axis. These axes intersect at a point called the origin, designated as (0, 0), dividing the plane into four sections known as quadrants.

• The first quadrant (Quadrant I) is where both x and y coordinates are positive.
• The second quadrant (Quadrant II) has negative x-coordinates and positive y-coordinates.
• The third quadrant (Quadrant III) contains points with both coordinates negative.
• The fourth quadrant (Quadrant IV) is where the x-coordinate is positive, and the y-coordinate is negative.

Understanding these basics is crucial for plotting points, understanding their relationships, and solving geometric problems using algebra.

Positive and Negative Coordinates

In coordinate geometry, the signs of a point’s coordinates are incredibly significant as they dictate the point's position relative to the origin.
A positive coordinate indicates that the point lies in the direction of increasing value along an axis starting from the origin, while a negative coordinate shows that the point is in the direction of decreasing value from the origin.

Interpreting Coordinates:

• A positive x-coordinate means the point is to the right of the y-axis.
• A negative x-coordinate means the point is to the left of the y-axis.
• A positive y-coordinate means the point is above the x-axis.
• A negative y-coordinate means the point is below the x-axis.

This understanding is vital for drawing graphs, solving equations graphically, and analyzing the spatial relationships between different geometric entities.

Quadrant Determination

Determining the quadrant in which a point lies is a fundamental skill in coordinate geometry. The quadrant is determined by the signs of the point's x and y coordinates.
Given a point with coordinates (\(h, -k\)) where \(h\) and \(k\) are both positive, we apply this logic:

• Since the x-coordinate (\(h\)) is positive, the point must lie to the right of the y-axis.
• Given that the y-coordinate (\(-k\)) is negative, the point is below the x-axis.

By combining these observations, we conclude that the point (\(h, -k\)) lies in Quadrant IV, which is characterized by positive x-coordinates and negative y-coordinates. Students should practice by considering different signs for coordinates and predicting the corresponding quadrant to enhance their proficiency in quadrant determination.