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.


10x² - 7y² = 140

Verified step by step guidance

1. Rewrite the given equation in the standard form of a conic section by dividing both sides by 140 to normalize it: \(\frac{10x^{2}}{140} - \frac{7y^{2}}{140} = 1\).
2. Simplify the fractions to get \(\frac{x^{2}}{14} - \frac{y^{2}}{20} = 1\).
3. Recognize the form of the equation: since it is of the form \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\), this represents a hyperbola centered at the origin with the transverse axis along the x-axis.
4. Identify the values \(a^{2} = 14\) and \(b^{2} = 20\). Use these to find the vertices at \((\pm a, 0)\), which are \((\pm \sqrt{14}, 0)\).
5. Calculate the foci using \(c^{2} = a^{2} + b^{2}\), so \(c = \sqrt{14 + 20} = \sqrt{34}\). The foci are at \((\pm c, 0)\). Then, find the equations of the asymptotes, which are \(y = \pm \frac{b}{a} x = \pm \frac{\sqrt{20}}{\sqrt{14}} x\).

Key Concepts

Identification of Conic Sections from General Equations

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The general second-degree equation Ax² + By² + Cx + Dy + E = 0 can represent a parabola, ellipse, or hyperbola depending on the signs and values of A and B. If A and B have the same sign and are unequal, the curve is an ellipse; if one is zero, it is a parabola; if they have opposite signs, it is a hyperbola.

Properties and Features of Parabolas, Ellipses, and Hyperbolas

Each conic section has unique geometric features: parabolas have a focus and directrix defining their shape; ellipses have two foci and vertices with major and minor axes; hyperbolas have two branches with vertices, foci, and asymptotes. Understanding these properties helps in labeling key points and sketching the graph accurately.

Graphing and Analyzing Conic Sections

To graph conics, rewrite the equation in standard form by completing the square if necessary. For ellipses and hyperbolas, calculate vertices, foci, and axes lengths or asymptote equations. For parabolas, find the focus and directrix from the standard form. This process enables precise plotting and understanding of the curve's shape.