Problem 1 - A Graphical Approach to Precalculus with Limits

Question

Match each equation in Column I with the appropriate description in Column II. Do not use a calculator. \(\mathbf{II}\) A. Hyperbola; center \((2,4)\) B. Ellipse; foci \((\pm 2 \sqrt{3}, 0)\) C. Hyperbola; foci \((0, \pm 2 \sqrt{5})\) D. Hyperbola; center \((-2,4)\) E. Ellipse; center \((-2,4)\) F. Center \((0,0) ;\) horizontal transverse axis G. Ellipse; foci \((0, \pm 2 \sqrt{3})\) H. Vertical major axis; center \((2,-4)\) $$\frac{x^{2}}{4}+\frac{y^{2}}{16}=1$$

Step-by-Step Solution

Answer

Equation matches with description G: Ellipse; foci \((0, \pm 2\sqrt{3})\).

Step 1: Identify the Equation Type

We see that the equation is of the form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This is the standard form of an ellipse centered at the origin (0,0). The presence of both \(x^2\) and \(y^2\) with positive signs indicates an ellipse.

Step 2: Determine the Axes

In the equation \(\frac{x^2}{4}+\frac{y^2}{16}=1\), \(a^2=4\) and \(b^2=16\). This implies that the axes along x and y are determined by \(a\) and \(b\) respectively. Since \(b > a\), this indicates a vertical major axis.

Step 3: Calculate the Foci Distance

For an ellipse, the foci are located at a distance \(c\) from the center along the major axis, where \(c^2 = b^2 - a^2\). Here, \(c^2 = 16 - 4 = 12\), so \(c = \sqrt{12} = 2\sqrt{3}\). The foci are (0, \(\pm 2\sqrt{3}\)).

Step 4: Match with Column II

The description that matches this information from Column II is G. Ellipse; foci \((0, \pm 2\sqrt{3})\). The identified features are consistent with the given options.

Key Concepts

Ellipses

Ellipses are a fundamental type of conic section. They occur in situations where a plane cuts through a cone at an angle that results in a closed, oval-shaped curve. One of the key characteristics of an ellipse is that it has two focal points, or foci. The interesting property of an ellipse is that the sum of the distances from any point on the ellipse to these two foci is constant.
This concept can help visualize how ellipses contrast with other conic sections like circles, which have only one center point. The general equation for an ellipse centered at (0,0) is given by:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]In this formula, \(a\) and \(b\) represent the distances from the center to the ellipse along the x-axis and y-axis, respectively. If \(b > a\), the ellipse is stretched more along the y-axis, giving it a vertical major axis, as in the original problem.

• Standard Form: \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] indicates an ellipse with a horizontal major axis when \(a > b\) or vertical major axis when \(b > a\).
• Main axes: The major and minor axes of an ellipse are the longest and shortest diameters respectively.

Understanding these basics will help you easily identify ellipses in math problems.

Hyperbolas

Hyperbolas, like ellipses, are one of the four fundamental conic sections. They appear when a plane intersects both nappes (the upper and lower halves) of a double cone in such a way that the resulting curve opens outward, unlike the closed shape of an ellipse.
One defining feature of hyperbolas is that they consist of two separate branches, unlike the continuous shape of an ellipse. The general equation for a hyperbola centered at (0,0) can be written as:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]or\[\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\]In these forms, the difference in signs indicates the hyperbola's unique open shape, where one axis measures the distance between the vertices while the other measures the distance between the foci.

• Center: The point halfway between the vertices.
• Axes: Each hyperbola has a transverse axis connecting the two vertices and a conjugate axis perpendicular to it.

Most importantly, unlike ellipses, the distance between the foci and any point on the hyperbola is constant when the distances to the branches are subtracted. This property is integral to the nature of hyperbolas.

Foci

Foci (plural of focus) are critical in understanding and solving problems involving conic sections like ellipses and hyperbolas. For any conic section shape, the foci play an important role in defining its geometric properties. In the case of ellipses, the foci are two distinct points such that the sum of the distances from any point on the ellipse to the foci is constant.
In hyperbolas, however, the nature of the foci differs. Here, the distance from any point on the curve to one focus, subtracted from the distance to the other focus, remains constant. This gives rise to the open shape characteristic of hyperbolas.
To calculate the distance to the foci, we use a slightly different approach for each shape:

• Ellipse Foci Calculation: For an ellipse, use \(c^2 = b^2 - a^2\) where the foci are along the major axis.
• Hyperbola Foci Calculation:For a hyperbola, use \(c^2 = a^2 + b^2\).

By comprehending these principles, you can solve problems involving foci effectively, recognizing their significant role in defining the nature and properties of conic sections.