Question

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

Accepted Answer

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}$$}{140} = 1. $$Simplify the fractions to get $$\frac{x$$^{2}$$}{14} - \frac{y$$^{2}$$}{20} = 1. $$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. Identify the values $$a^{2} = 14$ and b$$^{2}$$ = 20. $$Use these to find the vertices at $$(\pm a, 0)$$, which are $$(\pm \sqrt{14}, 0). $$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 $$y = \pm \frac{b}{a} x = \pm \frac{\sqrt{20}}{\sqrt{14}} x.$$

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Concetti chiave

Identification of Conic Sections from General Equations

Properties and Features of Parabolas, Ellipses, and Hyperbolas

Graphing and Analyzing Conic Sections