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.

x²/9 + y²/4 = 1

Accepted Answer

Identify the type of conic section by examining the given equation: $$\frac{x^2$}{9} + \frac{y^2$}{4} = 1. $$Since both $$x^2$$ and $$y^2$$ terms are positive and added together, this equation represents an ellipse. Recall the standard form of an ellipse centered at the origin: $$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} = 1$$, where $$a^2$$ and $$b^2$$ are the denominators under $$x^2$$ and $$y^2$$ respectively. Determine which is larger between $$a^2$$ and $$b^2 to $$identify the major and minor axes. Calculate the lengths of the major and minor axes. The length of the major axis is $$2a$$ and the length of the minor axis is $$2b. $$Here, $$a = \sqrt{9}$$ and $$b = \sqrt{4}. $$Find the coordinates of the vertices. If $$a^2 is $$under $$x^2$$, the major axis is along the x-axis, so vertices are at $$(\pm a, 0). If$$ $$a^2 is $$under $$y^2$$, the major axis is along the y-axis, so vertices are at $$(0, \pm a). $$Determine the foci using the formula $$c^2$$ = $$a^2$$ - $$b^2. $$Calculate $$c$$, then locate the foci at $$(\pm c, 0) if $$the major axis is horizontal, or at $$(0, \pm c) if $$vertical. Finally, sketch the ellipse labeling the vertices and foci accordingly.

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Concetti chiave

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Properties and Features of Ellipses

Graphing and Labeling Conic Sections