<< Chapter < Page | Chapter >> Page > |
What is the standard form equation of the ellipse that has vertices $\text{\hspace{0.17em}}\left(\mathrm{-3},3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,3\right)\text{\hspace{0.17em}}$ and foci $\text{\hspace{0.17em}}\left(1-2\sqrt{3},3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(1+2\sqrt{3},3\right)?$
$\frac{{\left(x-1\right)}^{2}}{16}+\frac{{\left(y-3\right)}^{2}}{4}=1$
Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form $\text{\hspace{0.17em}}\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,\text{}ab\text{\hspace{0.17em}}$ for horizontal ellipses and $\text{\hspace{0.17em}}\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{{a}^{2}}=1,\text{}ab\text{\hspace{0.17em}}$ for vertical ellipses.
Given the standard form of an equation for an ellipse centered at $\text{\hspace{0.17em}}\left(0,0\right),$ sketch the graph.
Graph the ellipse given by the equation, $\text{\hspace{0.17em}}\frac{{x}^{2}}{9}+\frac{{y}^{2}}{25}=1.\text{\hspace{0.17em}}$ Identify and label the center, vertices, co-vertices, and foci.
First, we determine the position of the major axis. Because $\text{\hspace{0.17em}}25>9,$ the major axis is on the y -axis. Therefore, the equation is in the form $\text{\hspace{0.17em}}\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{{a}^{2}}=1,$ where $\text{\hspace{0.17em}}{b}^{2}=9\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{a}^{2}=25.\text{\hspace{0.17em}}$ It follows that:
Therefore, the coordinates of the foci are $\text{\hspace{0.17em}}\left(\mathrm{0,}\pm 4\right).$
Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse. See [link] .
Graph the ellipse given by the equation $\text{\hspace{0.17em}}\frac{{x}^{2}}{36}+\frac{{y}^{2}}{4}=1.\text{\hspace{0.17em}}$ Identify and label the center, vertices, co-vertices, and foci.
center: $\text{\hspace{0.17em}}\left(0,0\right);\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(\pm 6,0\right);\text{\hspace{0.17em}}$ co-vertices: $\text{\hspace{0.17em}}\left(0,\pm 2\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(\pm 4\sqrt{2},0\right)$
Graph the ellipse given by the equation $\text{\hspace{0.17em}}4{x}^{2}+25{y}^{2}=100.\text{\hspace{0.17em}}$ Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.
First, use algebra to rewrite the equation in standard form.
Next, we determine the position of the major axis. Because $\text{\hspace{0.17em}}25>4,\text{\hspace{0.17em}}$ the major axis is on the x -axis. Therefore, the equation is in the form $\text{\hspace{0.17em}}\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}{a}^{2}=25\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{b}^{2}=4.\text{\hspace{0.17em}}$ It follows that:
Therefore the coordinates of the foci are $\text{\hspace{0.17em}}\left(\pm \sqrt{21},0\right).$
Next, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.
Notification Switch
Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?