Problem 50 - Algebra and Trigonometry

Question

Use your answers from Exercises 41–44 and the parametric equations given in Exercises 41–44 to find a set of parametric equations for the conic section or the line. Hyperbola: Vertices: (0, 4) and (0, -4); Foci: (0, 5) and (0, -5)

Step-by-Step Solution

Answer

The parametric equations for the given hyperbola are \(x = 3sin(t)\) and \(y = 4cos(t)\).

Step 1: Identifying the values of a and c

From the given information, the hyperbola's vertices are at (0, 4) and (0, -4), and its foci are at (0, 5) and (0, -5). The distance from the center of the hyperbola to a vertex is the value of 'a', and the distance to a focus is 'c'. Here, a = 4 and c = 5.

Step 2: Finding the value of b

Once we have the values of 'a' and 'c', we can use the relationship in hyperbolas that \(c^2 = a^2 + b^2\), to find the value of 'b'. Solving for b, using \(b = \sqrt{c^2 - a^2}\), gives \(b = \sqrt{5^2 - 4^2} = \sqrt{9} = 3\).

Step 3: Writing the equation of the hyperbola

The standard form of the equation of a vertical hyperbola centered at the origin is \(\frac {y^2} {a^2} - \frac {x^2} {b^2} = 1\). Substituting the values of a and b obtained, we get the equation of the hyperbola as \(\frac {y^2} {16} - \frac {x^2} {9} = 1\).

Step 4: Writing the parametric equations

For a hyperbola that opens up and down, the parametric equations are \(x = bsin(t)\) and \(y = acos(t)\). Substituting the values of a and b, we get the parametric equations as \(x = 3sin(t)\) and \(y = 4cos(t)\), where t is a parameter and will vary.

Key Concepts

Hyperbola

A hyperbola is a type of conic section, which is formed by the intersection of a plane and a cone. Unlike an ellipse, a hyperbola has two separate curves, which are also known as branches.
This is a significant aspect that distinguishes hyperbolas from other conic sections like ellipses and circles.

• The hyperbola has two foci. The foci provide an essential geometric property. For any point on the hyperbola, the difference in distances to the two foci is constant.
• Another important component of a hyperbola is its vertices. A vertex is where each branch is closest to the other, usually marking the turning point of the curve.

When working with hyperbolas, especially those centered at the origin, it is useful to use the distance between vertices and foci to derive more about its shape and orientation. In this exercise, we used the information from the vertices and foci to determine important parameters like 'a' and 'c,' essential for forming the equation of a hyperbola.

Conic Sections

Conic sections are the curves obtained when a plane intersects a double-napped cone. These shapes include the circle, ellipse, parabola, and hyperbola. They are called 'conic' because of their geometric origin from a cone.

• Circle: This occurs when the plane cuts the cone perpendicular to the axis. It's a special case of the ellipse where both foci are at the same point.
• Ellipse: An elongated circle occurring when the cutting plane is at an angle, but does not pass through the base of the cone.
• Parabola: Formed when the plane is parallel to the edge of the cone, resulting in a 'U'-shaped curve.
• Hyperbola: Occurs when the plane cuts through both nappes of the cone. It creates two separate curves that open either left-right or up-down.

Each type of conic section has its unique mathematical equations and properties. In problems involving conic sections, parametric equations can also be derived to represent their geometry conveniently.

Standard Form Equation

The standard form of an equation is often used to clearly express the geometry of a conic section. For hyperbolas, the standard form helps in understanding and plotting the curve on a coordinate plane.In vertical hyperbolas, which open upwards and downwards, the standard equation is \[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]Where:

• \(a\) is the distance from the center to each vertex.
• \(b\) is obtained using the relationship \(c^2 = a^2 + b^2\), where \(c\) is the distance to each focus.

This form of the equation reveals crucial information. Specifically, it tells how the hyperbola is oriented and the scale on each axis. This is essential when working with parametric equations, such as in the given exercise, to form a complete and clear mathematical model of the hyperbola. Understanding the standard form and the way it relates to 'a,' 'b,' and 'c' allows for accurate graphing and analysis of these fascinating curves.