Problem 37 - Precalculus with Limits

Question

In Exercises 33-40, use the results of Exercises 29-32 to find a set of parametric equations for the line or conic. Ellipse: vertices: \((\pm5, 0); \quad\) foci: \((\pm4, 0)\)

Step-by-Step Solution

Answer

The parametric equations for the given ellipse are \( \frac{x^2}{25}+ \frac{y^2}{9}=1 \)

Step 1: Identifying the Center

The center of the ellipse falls at the origin (0, 0) as the ellipse is symmetric with respect to the y-axis and x-axis.

Step 2: Calculating the value of 'a'

The distance from the center of the ellipse to each vertex is given to be 5 units. Thus, 'a' the semimajor axis length, is 5.

Step 3: Calculating the value of 'b'

The distance from the center to each focus is given to be 4 units i.e., 'c' is 4 units. Using the relation \( a^2 - b^2 = c^2 \), we can solve for 'b'. Substituting the values of 'a' and 'c', the equation becomes \( 5^2 - b^2 = 4^2 \). Solving this for 'b' we get 'b' as 3.

Step 4: Writing the Equation of the Ellipse

The equation of the ellipse is: \( \frac{x^2}{5^2}+ \frac{y^2}{3^2}=1 \), which simplifies to \( \frac{x^2}{25}+ \frac{y^2}{9}=1 \)

Key Concepts

Conic Sections

The term conic sections refers to the curves that result from the intersection of a plane with a double-napped cone. Depending on the angle and position of the plane cutting through the cone, the shapes formed can be circles, ellipses, parabolas, or hyperbolas.

Imagine slicing an orange (which represents the cone) at different angles: a horizontal cut gives us a circle, a tilted cut results in an ellipse, and more extreme angles produce parabolas or hyperbolas. Each of these shapes has distinct mathematical properties, making them important in a variety of scientific and engineering fields.

For instance, the orbits of planets are elliptical, satellite dishes often take a parabolic shape for better signal focus, and hyperbolas are essential in the construction of navigation systems like GPS.

Ellipse Characteristics

An ellipse is a conic section that resembles an elongated circle. It's defined by several key characteristics:

• Center: The fixed point that is equidistant from the 'vertices' (the ellipse's widest points).
• Vertices: These are the points on the ellipse that lie furthest from the center, forming the major axis.
• Foci (singular: focus): These are a pair of interior points such that the sum of the distances from any point on the ellipse to the foci is constant.
• Major axis: The longest line that can be drawn through the center and both vertices.
• Minor axis: The line perpendicular to the major axis that passes through the center, touching the ellipse on both sides.

The relationship between the distance to the foci and the vertices defines the shape of the ellipse. When the foci are closer together, the ellipse is more circular; when they are further apart, it becomes more elongated.

Moreover, the distances to the vertices (represented by 'a') and the foci (represented by 'c') follow a specific relationship with the distance to any point on the minor axis (represented by 'b'): \( a^2 = b^2 + c^2 \) for an ellipse centered at the origin, as in the given exercise.

Ellipse Standard Equation

The standard equation of an ellipse is a mathematical representation that allows us to visualize and analyze its characteristics. For an ellipse centered at the origin, the standard equation is:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

This equation arises from the definition of the ellipse as the set of points where the sum of the distances to the foci is constant. By substituting the values of \( a \) and \( b \), we can define the exact shape and size of a specific ellipse. For the ellipse in our exercise, with \( a = 5 \) and \( b = 3 \), the standard equation simplifies to:\[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]

Understandably, parametric equations give another way to represent an ellipse. They define a set of equations that express the coordinates of the points on the ellipse as functions of a parameter, usually denoted as \( t \). This can provide a more convenient form for plotting an ellipse or for solving problems in calculus and physics where a parameter is present.