Problem 12 - Algebra and Trigonometry

Question

In Exercises \(1-18,\) graph each ellipse and locate the foci. $$ y^{2}=1-4 x^{2} $$

Step-by-Step Solution

Answer

The given equation is actually not an ellipse and hence can't generate an ellipse graph with foci. The correct interpretation of the equation is a hyperbola, rather than an ellipse.

Step 1: Identify the type of conic section

The given equation \(y^{2} = 1 - 4x^{2}\) resembles the standard form of the equation of a horizontal ellipse, \(y^{2}/a^{2} - x^{2}/b^{2} = 1\). When you compare this standard form with the equation of the ellipse, you can conclude that \(a^{2} = 1\) and \(b^{2} = 1/4\).

Step 2: Find the semi-axes lengths (a and b)

From the equation, you found \(a^{2} = 1\), hence \(a=\sqrt{1} = 1\), the semi-minor axis. Similarly, for \(b^{2} = 1/4\), hence \(b = \sqrt{1/4} = 1/2\), the semi-major axis.

Step 3: Find the foci

Next, you can compute the coordinates of the foci using the equation \(c =\sqrt{ b^{2}-a^{2}}\). Since \ \(b^{2}\)=1/4 and \(a^{2}\)=1, it follows that \(c=\sqrt{1/4-1} = \sqrt{-3/4}\). Because c must be a real number, there is no real solution here. Thus, this is not an ellipse but a hyperbola.

Step 4: Correct the interpretation

Upon closer inspection, it appears that the initial analysis was incorrect: the given equation actually represents a hyperbola, not an ellipse. This discrepancy arose from the fact that for an ellipse, \(b^{2} > a^{2}\), while for a hyperbola, \(b^{2} < a^{2}\). In this case, the latter holds true, meaning the given equation is not an ellipse after all.

Key Concepts

Understanding Hyperbolas

A hyperbola is a fascinating type of conic section obtained by slicing a double cone, and it appears as two distinct and open curves. The standard form of a hyperbola is usually seen as \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). These equations may look similar to those of ellipses but are distinct in their placement of subtraction.

Hyperbolas have two branches, unlike the closed and single-loop shape of ellipses. The center of a hyperbola is the midpoint between its vertices, and this center is crucial for graphing the curves more easily. Another characteristic feature of a hyperbola is its asymptotes—lines that the branches of the hyperbola approach but never touch. These lines, intersecting at the center of the hyperbola, give the hyperbola its characteristic shape.

• **Vertex**: The point where each branch turns sharply.
• **Asymptote**: Guiding lines that define the direction and shape of the hyperbola.

Exploring Ellipses

An ellipse is another type of conic section and looks like an elongated circle. Its standard equation forms typically are \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) or \( \frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 \). Here, the plus sign plays a crucial role, as it distinguishes an ellipse from a hyperbola.

Ellipses have a beautiful symmetry. The numbers under \(x^2\) and \(y^2\), namely \(a^2\) and \(b^2\), determine the lengths of the major and minor axes. In most cases, \(a\) represents the semi-major axis and \(b\) the semi-minor axis, or vice versa, depending on which value is larger.

• **Axes**: Major and Minor, depending on value comparison \(a^2 > b^2\) or \(b^2 > a^2\).
• **Symmetry**: Mirror-like around its center.

Ellipses are especially notable in astronomy, as they describe the orbits of planets around the sun. This occurs because of the gravitational pull, creating a path that exemplifies the definition of an ellipse.

Foci in Conic Sections

The concept of foci (singular: focus) is central to understanding conic sections such as hyperbolas and ellipses. Foci are specific points that define the shape of the conic section. For an ellipse, any point on the ellipse maintains a constant total distance from the two foci. This property is what literally "shapes" an ellipse.

• For an ellipse, located on the major axis, inside the curve.
• For a hyperbola, located along the axis that passes through the vertices, outside the branches.

In a hyperbola, unlike an ellipse, the difference in distances from any point to the two foci remains constant. This difference leads to the open, dual-curved nature of hyperbolas, as opposed to the enclosed form of ellipses.

The understanding of foci also guides astronomers in interpreting paths and distances within celestial mechanics, making this concept not only important in mathematics but in real-world applications as well.