Graphing an ellipse with foci on the x-axis Sketch the graph and identify the foci of the ellipse x²/9 + y²/4 = 1.

Question

Graphing an ellipse with foci on the x-axis

Sketch the graph and identify the foci of the ellipse

[latex] \frac{x^{2}}{9}+\frac{y^{2}}{4}=1[/latex].

Accepted Answer

To sketch the ellipse, we find the x-intercepts and the y-intercepts. If x = 0, then [latex] y^{2} = 4[/latex] or [latex] y=\pm 2[/latex]. So the y-intercepts are (0, 2) and (0, -2). If y = 0, then [latex] x=\pm 3[/latex]. 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 [latex] c^{2}=a^{2}-b^{2}[/latex], to get [latex] c^{2}=9-4=5[/latex]. So [latex] c=\pm \sqrt{5}[/latex] , and the foci are [latex] (\sqrt{5}, 0)[/latex] and [latex] (-\sqrt{5}, 0)[/latex].

To check, solve for y to get [latex] y=\pm \sqrt{4-4 x^{2} / 9}[/latex]. Then graph [latex] y_{1}= \sqrt{4-4 x^{2} / 9}[/latex] and [latex] y_{2}=-y_{1}[/latex], as shown in Fig. 10.21.

Question 10.2.2: Graphing an ellipse with foci on the x-axis Sketch the graph...

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