Question

The point \((-2,3)\) lies in the quadrant : (a) First (b) Second (c) Third (d) Fourth

Short answer

Answer: (b) Second

Step-by-step solution

Recall the quadrants of the Cartesian plane

The Cartesian plane is divided into four quadrants, each distinguished by the signs of the x and y coordinates of the points within it. Starting from the top-right quadrant and going counter-clockwise, the quadrants are: 1. First quadrant: x > 0 and y > 0 2. Second quadrant: x < 0 and y > 0 3. Third quadrant: x < 0 and y < 0 4. Fourth quadrant: x > 0 and y < 0

Determine the quadrant for the given point

The point we're given is \((-2,3)\). Let's analyze the coordinates: - x-coordinate is \(-2\), which is negative - y-coordinate is \(3\), which is positive Looking back at the list of characteristics for each quadrant, we can see that a negative x-coordinate and a positive y-coordinate correspond to the second quadrant.

Choose the correct answer

Based on the analysis in Step 2, we can now determine that the point \((-2,3)\) lies in the second quadrant. Thus, the correct answer is: (b) Second

Key concepts

Coordinate Geometry

Understanding the basics of coordinate geometry is essential for analyzing the positions of points on a plane. This branch of mathematics combines algebra and geometry, enabling us to plot points, lines, and curves using an algebraic equation.

A point in this system is represented by a pair of numerical coordinates, which are the distances from the point to two fixed perpendicular directed lines, called axes. Consider an example: the point (4,5) has an x-coordinate (horizontal axis) of 4 and a y-coordinate (vertical axis) of 5. These coordinates help determine the precise location of a point within the Cartesian plane.

Coordinate geometry is not just about plotting points; it's also about understanding the relationship between the geometric figures and the algebraic equations that represent them. By doing so, we can solve geometric problems using algebraic methods such as finding the distance between two points, the midpoint of a line segment, or the equation of a line.

Cartesian Coordinate System

The Cartesian coordinate system, named after the French mathematician René Descartes, is a two-dimensional plane with two perpendicular lines or axes that intersect at a point called the origin. The horizontal axis is known as the x-axis, and the vertical axis is called the y-axis.

To plot a point on this plane, we use an ordered pair of numbers \(x, y\), where 'x' represents the horizontal position and 'y' represents the vertical position relative to the origin. Each number in the pair is called a 'coordinate'. The position of the point \(x, y\) is found by starting at the origin, moving 'x' units along the x-axis, and then 'y' units parallel to the y-axis.

The intersection of the axes divides the plane into four equal sections known as quadrants. The system allows us to precisely determine locations and describe geometric relationships algebraically, which is the foundation for more advanced topics such as calculus and linear algebra.

Quadrant Analysis

In the Cartesian coordinate system, quadrant analysis is the process of determining which of the four quadrants a point belongs to. Each quadrant corresponds to a unique combination of signs for the x and y coordinates.

Here is the breakdown:

• First Quadrant: Both x and y are positive.
• Second Quadrant: x is negative, y is positive.
• Third Quadrant: Both x and y are negative.
• Fourth Quadrant: x is positive, y is negative.

By analyzing the signs of a point's coordinates, we can quickly identify its quadrant. For example, a point with coordinates \(3, -4\) lies in the fourth quadrant because the x-value is positive and the y-value is negative. This type of analysis is fundamental for graphing equations and understanding the spatial relationships between different geometrical entities on the plane.