13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.


x² + y²/9 = 1

Verified step by step guidance

1. Identify the type of conic section by comparing the given equation to standard forms. The equation is \(x^{2} + \frac{y^{2}}{9} = 1\). Since both \(x^{2}\) and \(y^{2}\) terms are positive and the equation equals 1, this suggests it is an ellipse.
2. Rewrite the equation in the standard form of an ellipse: \(\frac{x^{2}}{1} + \frac{y^{2}}{9} = 1\). Here, \(a^{2} = 9\) and \(b^{2} = 1\), where \(a^{2}\) is the larger denominator corresponding to the major axis.
3. Determine the lengths of the major and minor axes. The major axis length is \(2a = 2 \times 3 = 6\), and the minor axis length is \(2b = 2 \times 1 = 2\).
4. Find the coordinates of the vertices. Since \(a^{2} = 9\) is under \(y^{2}\), the major axis is vertical. The vertices are at \((0, \pm a) = (0, \pm 3)\).
5. Calculate the foci using \(c^{2} = a^{2} - b^{2}\). Here, \(c^{2} = 9 - 1 = 8\), so \(c = \sqrt{8} = 2\sqrt{2}\). The foci are located at \((0, \pm c) = (0, \pm 2\sqrt{2})\).

Key Concepts

Identification of Conic Sections

Conic sections are curves obtained by intersecting a plane with a cone, resulting in parabolas, ellipses, or hyperbolas. The general form of their equations helps classify them: ellipses have sums of squared terms equal to 1 with positive coefficients, hyperbolas have a difference of squared terms, and parabolas have one squared term. Recognizing these forms is essential to determine the type of conic.

Properties and Features of Ellipses

An ellipse is defined by two foci such that the sum of distances from any point on the ellipse to the foci is constant. Key features include vertices (endpoints of the major axis), foci, and the lengths of the major and minor axes. Understanding these properties allows for accurate graphing and labeling of the ellipse.

Graphing and Analyzing Conic Sections

Graphing conic sections involves plotting key points like vertices and foci, and drawing axes or asymptotes where applicable. For ellipses, labeling the major and minor axes is crucial, while hyperbolas require asymptotes. This process helps visualize the curve and understand its geometric behavior.