Question 10.2.2: Graphing an ellipse with foci on the x-axis Sketch the graph...
Graphing an ellipse with foci on the x-axis
Sketch the graph and identify the foci of the ellipse
\frac{x^{2}}{9}+\frac{y^{2}}{4}=1.
To sketch the ellipse, we find the x-intercepts and the y-intercepts. If x = 0, then y^{2} = 4 or y=\pm 2. So the y-intercepts are (0, 2) and (0, -2). If y = 0, then x=\pm 3. So the x-intercepts are (3, 0) and (-3, 0). To make a rough sketch of an ellipse, plot only the intercepts and draw an ellipse through them, as shown in Fig. 10.20. Since this ellipse is elongated in the direction of the x-axis, the foci are on the x-axis. Use a = 3 and b = 2 in c^{2}=a^{2}-b^{2}, to get c^{2}=9-4=5. So c=\pm \sqrt{5} , and the foci are (\sqrt{5}, 0) and (-\sqrt{5}, 0).
To check, solve for y to get y=\pm \sqrt{4-4 x^{2} / 9}. Then graph y_{1}= \sqrt{4-4 x^{2} / 9} and y_{2}=-y_{1}, as shown in Fig. 10.21.
