91Ó°ÊÓ

Divide the given equation by 36 to get the ellipse in the standard form: \[ \frac{x^2}{9} + \frac{y^2}{4} = 1. \]

From the standard equation \(\frac{x^2}{9} + \frac{y^2}{4} = 1\), we can see that the length of the major and minor axes are given by: Major axis: \(2a = \sqrt{9} \times 2 = 6\) Minor axis: \(2b = \sqrt{4} \times 2 = 4\) Thus, a = 3 and b = 2.

To find the distance between the foci, we need to calculate the value of c using the relationship between the semi-major axis (a), the semi-minor axis (b), and the distance between the foci (c): \[ c = \sqrt{a^2 - b^2} = \sqrt{3^2 - 2^2} = \sqrt{9-4} = \sqrt{5} \] The foci are located \(c = \sqrt{5}\) units from the center of the ellipse along the major axis.

Using the values of a, b, and c, we can determine the foci and vertices of the ellipse: Foci: \((\pm\sqrt{5}, 0)\) Vertices: \((\pm 3, 0)\) along the major axis, \((0, \pm 2)\) along the minor axis.

To sketch the graph of the ellipse, use the following steps: 1. Draw the major and minor axes on the coordinate plane. 2. Plot the center of the ellipse at the origin (0, 0). 3. Plot the foci at points \((\pm\sqrt{5}, 0)\) on the major axis. 4. Plot the vertices at points \(\pm 3, 0\) on the major axis, and \((0, \pm2)\) on the minor axis. 5. Draw the ellipse by connecting the vertices with a smooth curve, making sure it passes through the vertices and gets closer to the foci as it approaches the major axis. By following these steps, we now have the graph of the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\), with the foci \((\pm\sqrt{5}, 0)\), vertices \((\pm 3, 0) \) along the major axis and \((0, \pm 2)\) along the minor axis.