Question

Graph each ellipse and locate the foci. x2/(9/4) +y2/(25/4) = 1

Accepted Answer

Identify the standard form of the ellipse equation given: $$\frac{x$$^{2}$$}{\frac{9}{4}} + \frac{y$$^{2}$$}{\frac{25}{4}} = 1. $$This matches the form $$\frac{x$$^{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 denominator is larger to identify the major axis. Compare $$a^{2}$$ = \frac{9}{4}$$ and b$$^{2}$$ = \frac{25}{4}. $$The larger value corresponds to the major axis, which tells us if the ellipse is vertical or horizontal. Calculate the lengths of the semi-major axis $$a$$ and semi-minor axis $$b by $$taking the square roots: $$a = \sqrt{	ext{larger denominator}}$$ and $$b = \sqrt{	ext{smaller denominator}}. $$Find the distance $$c$$ from the center to each focus using the relationship $$c^{2}$$ = $$a^{2}$$ - $$b^{2}. $$Compute $$c by $$taking the square root of $$c^{2}. $$Locate the foci on the coordinate plane along the major axis at points $$(0, \pm c) if $$the major axis is vertical, or $$(\pm c, 0) if $$horizontal. Then sketch the ellipse using the intercepts $$\pm a$$ and $$\pm b on $$the axes.

Graph each ellipse and locate the foci. x²/(9/4) +y²/(25/4) = 1

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Key Concepts

Standard Form of an Ellipse

Graphing an Ellipse

Locating the Foci of an Ellipse