Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, find the foci for the given ellipses. \(\frac{(x+1)^{2}}{100}+\frac{(y-2)^{2}}{4}=1\)

Step-by-Step Solution

Answer

The foci are at \((-1 + 4\sqrt{6}, 2)\) and \((-1 - 4\sqrt{6}, 2)\).

Step 1: Identify the Standard Form

The given equation is \( \frac{(x+1)^{2}}{100} + \frac{(y-2)^{2}}{4} = 1 \). This is in the standard form of an ellipse \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \). Here, \((h,k) = (-1,2)\) is the center of the ellipse, \(a^2 = 100\), and \(b^2 = 4\).

Step 2: Determine the Major and Minor Axes

Compare \(a^2\) and \(b^2\) to identify the major axis. Since \(a^2 = 100 > 4 = b^2\), the major axis is horizontal. Thus, \(a = \sqrt{100} = 10\) and \(b = \sqrt{4} = 2\).

Step 3: Calculate the Distance to the Foci

For an ellipse, the distance from the center to each focus along the major axis is given by \(c = \sqrt{a^2 - b^2}\). Substituting the given values: \[c = \sqrt{100 - 4} = \sqrt{96} = 4\sqrt{6}.\]

Step 4: Find the Coordinates of the Foci

Since the major axis is horizontal, the foci are located at \((h \pm c, k)\). Substitute the values: \((-1 \pm 4\sqrt{6}, 2)\). Thus, the foci are \((-1 + 4\sqrt{6}, 2)\) and \((-1 - 4\sqrt{6}, 2)\).

Key Concepts

Conic Sections

Conic sections are shapes created by slicing a right circular cone with a plane. The resulting curves are highly significant in mathematics and include ellipses, parabolas, hyperbolas, and circles. Each shape has unique properties and occurs under specific conditions, mainly dictated by the angle and position of the intersecting plane relative to the cone.

An ellipse, specifically, is formed when the plane cuts through the cone at an angle less than that of the cone's side but does not pass through the apex. This construction can result in either a single closed curve or two separate curves, depending on the cone's properties. The diagrammatic representation of conic sections allows for the exploration of geometric properties, which often leads to exciting applications in physics, astronomy, and engineering.

To understand conic sections better, it is helpful to compare them through their equations and geometric properties:

• Ellipses: Symmetrical along two axes and form a closed curve.
• Parabolas: Have an open curve shape with a single axis of symmetry.
• Hyperbolas: Comprise two open curves with two axes of symmetry.
• Circles: Are special ellipses where both axes are of equal length.

Understanding these elementary aspects of conic sections is crucial for exploring complex spatial relationships, especially in the study of ellipses.

Standard Form of an Ellipse

To effectively work with ellipses, knowing their standard form equations is essential. The standard form helps determine the properties of the ellipse, like its size, shape, and orientation. This understanding and equation manipulation are critical in solving geometry problems effectively.

The standard form of an ellipse with a horizontal major axis is:\[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]
Here's what each component represents:

• (h, k): The center of the ellipse. It shifts the ellipse from the origin to any point in the coordinate plane based on the values of \(h\) and \(k\).
• a: Represents the distance from the center of the ellipse to the edge along the major axis. The variable \(a\) is always the larger value if the major axis is horizontal.
• b: Represents the distance from the center to the edge along the minor axis.

A vertical major axis ellipse has its axes swapped in the equation:\[\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\]
Whenever you come across an equation in this format, identifying the components can help you sketch and analyze the ellipse's properties with precision.

Foci of an Ellipse

The foci of an ellipse, known singularly as a focus, are two distinct points within the ellipse that help define its shape. The total distance from any point on the ellipse to the two foci is a constant, making the foci fundamental in understanding an ellipse's geometric properties.

To derive the coordinates of the foci in an ellipse:

• First, identify whether the major axis is horizontal or vertical.
• Calculate the value of \(c\), where \(c = \sqrt{a^2 - b^2}\).
• For a horizontal major axis, the foci are located at \((h \pm c, k)\).
• For a vertical major axis, the foci would be at \((h, k \pm c)\).

In our given problem, the foci are critical in confirming the ellipse's configuration and solving it. By calculating \(c\) and applying it in the equation, you obtain the precise coordinates of the foci.This constant sum of distances creates the unique elliptic shape as you draw the curve around and through these focal points.

Recognizing and understanding the role of ellipses' foci can illuminate various natural and theoretical phenomena, aiding studies in astronomy, optics, and even planetary orbits.