Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the given central conic. Ellipse with a focus at \((-3,0)\) and a vertex at \((6,0)\)

Step-by-Step Solution

Answer

The conditions describe a degenerate ellipse where \( b = 0 \), suggesting a line or point.

Step 1: Determine the orientation

Since the focus and vertex are on the x-axis, the ellipse is horizontally oriented.

Step 2: Identify the center

The center of the ellipse is the midpoint between the vertex and the corresponding focus. Therefore, the center is at \( (\frac{6 + (-3)}{2}, 0) = (1.5, 0) \).

Step 3: Calculate \(c\)

The distance \(c\) from the center to the focus is \( |-3 - 1.5| = 4.5 \).

Step 4: Calculate \( a \)

The distance \(a\) from the center to the vertex is \( |6 - 1.5| = 4.5 \).

Step 5: Use the relationship \(c^2 = a^2 - b^2\)

For an ellipse, \( c^2 = a^2 - b^2 \). Here, \(a = 4.5\), so \(a^2 = 20.25\). Given \(c = 4.5\), then \(c^2 = 20.25\). Thus, \( 20.25 = 20.25 - b^2 \), so \(b^2 = 0\).

Step 6: Construct the ellipse equation

Since \( b^2 = 0 \), this is actually a degenerate case usually representing a line or point, but not a proper ellipse. Normally, the equation in standard form is \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), so if \(b eq 0\), the parameters indicate that geometrically it converges.

Key Concepts

Understanding Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone. These include circles, ellipses, parabolas, and hyperbolas. They are used in various fields such as astronomy and physics because they describe the paths followed by celestial bodies.

An ellipse is one such conic section and has a unique shape, often compared to an elongated circle.In the coordinate plane, the equation of an ellipse can be written as:

• \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\\]

where \((h, k)\) is the center, \(a\) and \(b\) are the semi major and semi minor axes respectively.

By analyzing the positions of the focus and vertex point given in a problem, we understand the properties of the ellipse and construct its equation. This is crucial because the position and orientation of the conic section define its unique mathematical representation.

Focusing on Focus and Vertex

Ellipses have two foci, and the sum of the distances from any point on the ellipse to the two foci is constant. The focus is crucial in dictating the shape of the ellipse. In our exercise, the focus \((-3,0)\) lies on the x-axis.

The vertex is one of the two points where the ellipse intersects its major axis. In our problem, the vertex is at \((6,0)\).

To find the center, it's important to locate it directly between the focus and the vertex. With a focus at \((-3,0)\) and vertex at \((6,0)\), the midpoint gives us the center at\(((6 + (-3)) / 2, 0) = (1.5, 0)\).

This understanding of the focus and vertex aids in defining the ellipse's essential characteristics.

Locating the Ellipse Center

The center of an ellipse plays a significant role in its geometric configuration. Knowing the center allows us to define the symmetry and orientation of the ellipse.

In a plane, the center of an ellipse is found exactly halfway between its foci.For ellipses centered around the x-axis, like in our example, the x-coordinate of the center is calculated by averaging the x-coordinates of the vertex and its corresponding focus.

• Here, the center is at \((1.5, 0)\), as calculated from the midpoint between \((-3,0)\) and \((6,0)\).

The center forms part of the standard equation of an ellipse, contributing to the terms\((x-h)^2\) and\((y-k)^2\) in its equation.

Understanding the center helps map the overall orientation and helps easily transform the ellipse in translation problems.

Exploring Ellipse Orientation

The orientation of an ellipse determines the axis along which it is elongated, and this affects how the ellipse will look on a graph.

There are two main orientations based on the position of the foci:

• Horizontal Orientation: When the foci and vertices lie along the x-axis. An example is the case in our exercise.
• Vertical Orientation: When the foci and vertices lie along the y-axis.

For our problem, because both the focus\((-3,0)\) and the vertex\((6,0)\) are on the x-axis, the ellipse is horizontally oriented.

This means that the longer axis, the major axis, runs parallel to the x-axis, impacting the value of\(a\) and determining the ellipse's layout.Knowing an ellipse's orientation helps choose the correct form of its equation and properly interpret geometric problems.