Problem 51 - Algebra 2

Question

The graph of which ellipse contains all the points in the table below? $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {-4} & {-2} & {0} & {2} & {4} \\\ \hline y & {0} & { \pm \sqrt{3}} & { \pm 2} & { \pm \sqrt{3}} & {0} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

The graph of the ellipse that contains all the provided points is described by the equation \( x^2/16 + y^2/4 = 1 \)

Step 1: Identify the values of a and b

From the given points, one can notice that the x-coordinate ranges from -4 to 4 and the y-coordinate ranges from -2 to 2. This suggests that the larger axis is 'a' is 4, and the smaller axis 'b' is 2.

Step 2: Formulate the standard equation

Using the derived values of a and b, one gets the standard equation for the ellipse: \( (x^2 / (4)^2) + (y^2 / (2)^2) = 1 \), which simplifies to \( x^2/16 + y^2/4 = 1 \).

Step 3: Verify the equation with the points

To verify that this equation fits all the points provided in the table, substitute each pair of x and y values into the equation. For example, when x = -4, y = 0, substituting these into the equation gives: \( (-4)^2/16 + (0)^2/4 = 1 \), which is true. Do the same for other points to verify. It should be clear that all points fit into the equation.

Key Concepts

Conic Sections

Conic sections are fundamental shapes obtained by intersecting a plane with a cone. These shapes include circles, ellipses, parabolas, and hyperbolas. They play an important role in mathematics due to their unique properties and applications in various fields.
Circles are special cases of ellipses, where the two foci coincide, producing a perfectly symmetrical figure. Parabolas are often used to model the path of objects under the influence of gravity, and hyperbolas appear in scenarios involving certain types of acceleration.

• Different conic sections are determined by the angle at which the plane intersects the cone.
• The resulting shapes have distinctive equations and geometric properties.

Ellipses, like the one in your exercise, have a distinct oval shape and are defined by their foci and axes lengths. Understanding conic sections is crucial for solving problems involving these unique geometric figures.

Standard Equation of Ellipse

The standard equation of an ellipse is a key tool in coordinate geometry for describing the ellipse mathematically. In its simplest form, the equation is written as:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Here, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis. These values correspond to half of the ellipse's longest and shortest widths, respectively.
The standard equation can vary slightly depending on whether the ellipse is oriented horizontally or vertically; in our exercise, it's horizontal. This mathematical representation allows us to predict the location and shape of the ellipse and understand properties such as symmetry and the distances from the center to the foci.

• The center of the ellipse in standard form is at the origin, \((0, 0)\).
• The axes lengths determine the spread and orientation of the ellipse.

This equation helps confirm whether a set of points belongs to a given ellipse.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, offers a powerful way to investigate geometric properties using algebra. In this realm, we explore ellipses entirely within the coordinate plane, bringing together algebraic equations and geometric constructs.
Using coordinate geometry, you can determine positions, calculate distances, and verify forms like ellipses by plugging coordinates into equations. This approach simplifies the analysis of various shapes and sizes of geometric figures.

• It connects algebra with geometry, providing a deeper understanding of both fields.
• This duality is particularly handy for solving problems involving curves and distances, as seen with ellipses.

By applying coordinate geometry, you can predict the behavior of ellipses, such as their orientation, direction, and intercepts within the coordinate plane.

Graphing Ellipses

Graphing ellipses provides a visual representation of their mathematical structure. When graphing, the standard equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) is your starting point. This formula translates into a graph by plotting points that satisfy the equation.
You will identify the key elements: the center of the ellipse, the lengths of the semi-major (\(a\)) and semi-minor (\(b\)) axes, and the position of the foci. The orientation of these elements determines the exact shape and alignment of the ellipse in the plane.

• Start at the center, plot the lengths of the axes along the x- and y-axis, and trace the ellipse's curvature.
• Make sure the plotted points satisfy the initial equation, ensuring accuracy.

Visualizing ellipses through graphing provides a tangible understanding of their geometrical and algebraic properties. This method allows for easy verification against given data points, as illustrated in your exercise.