Question

Given the equation: \(\mathrm{x}^{2}-\mathrm{y}^{2}=4\), describe what inequalities hold in the regions of the plane which this curve separates.

Short answer

The given equation represents a hyperbola. For points that lie inside the hyperbola, the inequality is \(x^2 - y^2 < 4\). For points that lie outside the hyperbola, the inequality is \(x^2 - y^2 > 4\). These inequalities describe the regions of the plane that the curve separates.

Step-by-step solution

1. Rewrite the equation in standard form

The given equation is in the following form: \(x^2 - y^2 = 4\). Divide both sides by 4 to get it in the standard form: \[\frac{x^2}{(2)^2} - \frac{y^2}{(2)^2} = 1\]

2. Analyze points inside and outside of the hyperbola

Let's analyze a point (a, b) where \(a^2 - b^2 < 4\) to see which inequality (either < or >) corresponds to the region inside or outside the hyperbola. Let \(a = 0\) and \(b = 0\), so we have \(a^2 - b^2 = 0\), and since \(0 < 4\), this point lies inside the hyperbola. Thus, points that have the inequality \(a^2 - b^2 < 4\) are inside the hyperbola. Similarly, we can analyze a point (c, d) where \(c^2 - d^2 > 4\). Let \(c = 3\) and \(d = 0\), so we get \(c^2 - d^2 = 9 > 4\), and this point lies outside the hyperbola. Consequently, points that have the inequality \(c^2 - d^2 > 4\) lie outside the hyperbola.

3. Rewrite inequalities in terms of x and y

Finally, we rewrite the inequalities using x and y instead of a, b, c, and d. For points that lie inside the hyperbola, the inequality is: \[x^2 - y^2 < 4\] For points that lie outside the hyperbola, the inequality is: \[x^2 - y^2 > 4\] These inequalities describe the regions of the plane that the curve separates.

Key concepts

Inequalities

When we talk about inequalities in the context of hyperbolas, it's all about understanding where certain points lie relative to the hyperbola on a plane. A hyperbola itself is a type of conic section, a curve which can be represented by the equation \(x^2 - y^2 = 4\) in our case. This equation defines the actual hyperbola, but inequalities help us describe other regions relative to it.

• If a point (x, y) satisfies \(x^2 - y^2 < 4\), it means the point is inside the hyperbola.
• Conversely, if \(x^2 - y^2 > 4\), the point lies outside the hyperbola.
• When \(x^2 - y^2 = 4\), the point is on the hyperbola itself.

These inequalities give us a clearer picture of how the hyperbola divides the plane into distinct regions, allowing us to categorize points in terms of their closeness or distance from the curve.

Standard Form

Standard form helps in easily identifying and understanding the main features of different conic sections. To rewrite an equation in standard form is like putting it into a special structure which highlights these features.

In the context of hyperbolas, the standard form can simplify how we view them. The given equation is \(x^2 - y^2 = 4\). By dividing everything by 4, we can turn it into its standard form: \[\frac{x^2}{(2)^2} - \frac{y^2}{(2)^2} = 1\]This form reveals that the center of the hyperbola is at the origin (0,0), and the distances along the x-axis and y-axis, called the lengths of the semi-major and semi-minor axes, are both 2 units. This transformation helps in identifying how and where the hyperbola opens and allows us to easily plot or sketch it.

Regions of the Plane

Understanding the regions of the plane is crucial when dealing with non-linear equations, like those of hyperbolas. Every hyperbola splits the plane into three distinct regions:

• The interior region: This is where points satisfy the inequality \(x^2 - y^2 < 4\). Here, the points are nestled within the arms of the hyperbola.
• The exterior region: Points here satisfy the inequality \(x^2 - y^2 > 4\). They lie outside the curve, at a distance greater than the given.
• The boundary: Points precisely on the hyperbola satisfy \(x^2 - y^2 = 4\). They are the exact points through which the hyperbola passes.

By understanding these regions, you can analyze and solve problems related to the placement of points relative to the defined curves, which is important in both pure math and applied sciences.

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. These include circles, ellipses, parabolas, and hyperbolas. Each type has its unique properties and equations.

Hyperbolas, like the one described by the equation \(x^2 - y^2 = 4\), have two symmetrical branches which curve away from each other, resembling an open pair of tails. Understanding hyperbolas entails recognizing their standard form, their opening direction, and their asymptotes - the lines they approach but never touch.

• Standard form helps tell apart hyperbolas from other conic sections.
• The equation in standard form normally looks like \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) for hyperbolas.
By appreciating these features and their distinctions from other conics, students can gain insight into the broader scope of algebra and geometry.