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How to plot a region correctly

In JEE Mains and Advanced problems, especially when it comes to calculating areas bounded by curves, one of the initial challenges students often face is correctly plotting the region in question. This can be particularly true for complex shapes or when dealing with inequalities that define regions not immediately intuitive to visualize. If you find yourself grappling with how to plot these regions effectively, you're not alone. The following discussion aims to shed light on a systematic approach that can simplify this process, making it not just manageable but also insightful.

First we need to understand what a region is:

Region

A region in two dimensions is a subset of the plane characterized by certain conditions on the coordinates of its points. When a curve is defined by an equation \(f(x, y) = 0\), where \(f(x, y)\) is a mathematical expression involving \(x\) and \(y\), it naturally divides the plane into distinct regions based on the sign of \(f(x, y)\).

  1. The region where \(f(x, y) > 0\) consists of all points \((x, y)\) for which the value of \(f(x, y)\) is positive. This set of points does not include those on the curve itself but lies on one side of it, depending on the nature of the curve.

  2. Conversely, the region where \(f(x, y) < 0\) comprises all points \((x, y)\) for which the value of \(f(x, y)\) is negative. These points occupy the other side of the curve relative to the region where \(f(x, y) > 0\).

The curve \(f(x, y) = 0\) acts as a boundary separating these two regions. Points on the curve itself neither belong to the region where \(f(x, y) > 0\) nor to the region where \(f(x, y) < 0\).

How a curve divides a region

Let's consider two examples to illustrate the concept of regions divided by a curve:

Example 1: Circle Defined by \(f(x, y) = x^2 + y^2 - r^2\)

For a circle centered at the origin with radius \(r\), the defining equation is \(f(x, y) = x^2 + y^2 - r^2 = 0\).

  • Region where \(f(x, y) > 0\): This corresponds to the set of points \((x, y)\) for which \(x^2 + y^2 - r^2 > 0\), or equivalently, \(x^2 + y^2 > r^2\). These points lie outside the circle, as their distance from the origin is greater than \(r\).

  • Region where \(f(x, y) < 0\): This is the set of points \((x, y)\) for which \(x^2 + y^2 - r^2 < 0\), or \(x^2 + y^2 < r^2\). These points are located inside the circle, since their distance from the origin is less than \(r\).

Showing regions with respect to a circle

Example 2: Parabola Defined by \(f(x, y) = y - x^2\)

For a parabola that opens upwards, the relationship between \(x\) and \(y\) can be described by \(f(x, y) = y - x^2 = 0\).

  • Region where \(f(x, y) > 0\): This region consists of points \((x, y)\) satisfying \(y - x^2 > 0\), or \(y > x^2\). These points lie above the parabola, as their \(y\)-coordinate is greater than the square of their \(x\)-coordinate.

  • Region where \(f(x, y) < 0\): This includes points \((x, y)\) for which \(y - x^2 < 0\), or \(y < x^2\). These points are found below the parabola, indicating a \(y\)-coordinate that is less than the square of their \(x\)-coordinate.

Showing regions of a parabola

In both examples, the curve \(f(x, y) = 0\) serves as a boundary that distinctly separates the plane into two regions, demonstrating the nature of the relationship between \(x\) and \(y\) in defining geometric shapes and areas in two dimensions.

Plotting

Suppose we wish to plot the region determined by the condition \(f(x, y) > 0\), where \(f(x, y)\) is a given mathematical expression relating \(x\) and \(y\), execute the following procedure:

  1. Sketch the Boundary: The boundary of the region is defined by the curve \(f(x, y) = 0\). Start by accurately plotting this curve on a Cartesian plane. This involves identifying points \((x, y)\) that satisfy the equation and connecting these points smoothly to form the curve.

  2. Identify Test Points: After sketching the boundary, select a test point not on the curve to determine which side of the boundary satisfies the condition \(f(x, y) > 0\). The test point should be chosen for ease of calculation, such as a point where either \(x\) or \(y\) is zero, if applicable.

  3. Evaluate \(f(x, y)\) at the Test Point: Substitute the coordinates of the test point into the expression \(f(x, y)\) and evaluate. The sign of the result will indicate whether the test point lies in the region where \(f(x, y) > 0\) or not.

    • If \(f(x, y) > 0\) at the test point, then the region containing this point satisfies the given condition. This area is part of the desired region to be plotted.

    • If \(f(x, y) < 0\), then the opposite side of the boundary defined by \(f(x, y) = 0\) contains the points of interest.

  4. Shade the Region: Based on the outcome of evaluating the test point, shade the area of the plane that meets the condition \(f(x, y) > 0\). This visually represents all points \((x, y)\) that satisfy the inequality.