Understanding Quadrants in Coordinate Geometry

The question asks us to identify the quadrants in which two given points, $(-2, 3)$ and $(3, -2)$, are located. The Cartesian coordinate system is divided into four quadrants based on the signs of the x and y coordinates.

What are Quadrants?

The x-axis and y-axis divide the plane into four regions called quadrants. They are numbered counterclockwise, starting from the top-right region.

• Quadrant I: Both x and y coordinates are positive. (x > 0, y > 0)
• Quadrant II: The x-coordinate is negative, and the y-coordinate is positive. (x < 0, y > 0)
• Quadrant III: Both x and y coordinates are negative. (x < 0, y < 0)
• Quadrant IV: The x-coordinate is positive, and the y-coordinate is negative. (x > 0, y < 0)

Analyzing Point 1: $(-2, 3)$

Let's look at the coordinates of the first point, $(-2, 3)$.

• The x-coordinate is $-2$. This is a negative value (x < 0).
• The y-coordinate is $3$. This is a positive value (y > 0).

The signs of the coordinates are negative for x and positive for y (-, +). According to the rules for quadrants, a point with a negative x-coordinate and a positive y-coordinate lies in the Second Quadrant.

Analyzing Point 2: $(3, -2)$

Now let's look at the coordinates of the second point, $(3, -2)$.

• The x-coordinate is $3$. This is a positive value (x > 0).
• The y-coordinate is $-2$. This is a negative value (y < 0).

The signs of the coordinates are positive for x and negative for y (+, -). According to the rules for quadrants, a point with a positive x-coordinate and a negative y-coordinate lies in the Fourth Quadrant.

Conclusion

Based on our analysis:

• The point $(-2, 3)$ lies in the Second Quadrant.
• The point $(3, -2)$ lies in the Fourth Quadrant.

Therefore, the points $(-2, 3)$ and $(3, -2)$ lie in the Second and Fourth quadrants, respectively.

Revision Table: Quadrant Rules Summary

Additional Information: Plotting Points and Axes

Points that lie on the axes do not belong to any quadrant.

• If the x-coordinate is $0$ (e.g., $(0, 5)$ or $(0, -3)$), the point lies on the y-axis.
• If the y-coordinate is $0$ (e.g., $(2, 0)$ or $(-4, 0)$), the point lies on the x-axis.
• The origin $(0, 0)$ is the intersection of the x and y axes and is also not in any quadrant.

Understanding how to plot points and determine their location in the Cartesian plane is fundamental in coordinate geometry.