Question

Find the center, the vertices, the foci, and the asymptotes. Then draw the graph. {x² / 9 }​ + {y² / 29 ​= 1} The vertices are ______

Answer 1

To find the center of the given ellipse, use the coordinates from the equation. The vertices can be found using the square root of the numbers in the equation. The foci can be found using a formula. The asymptotes can be determined using the coefficients in the equation. Finally, graph the ellipse using these values.

Step-by-step explanation:

An ellipse is defined by its center, vertices, foci, and asymptotes. To find the center of the ellipse, we simply take the coordinates of the x and y terms in the equation. For the given equation, the center is (0, 0). The vertices of the ellipse can be found by taking the square root of 9 and 29 in the equation. So the vertices are (3, 0) and (-3, 0). The foci can be found using the formula c = sqrt(a^2 - b^2), where a is the bigger number and b is the smaller number in the equation. So the foci are (0, 5.39) and (0, -5.39). The asymptotes for the given ellipse are the lines y = (29/9)x and y = -(29/9)x.

To graph the ellipse, plot the center, the vertices, and the foci. Then draw the asymptotes and sketch the curve connecting the vertices. The resulting graph will be an ellipse.

Answer 2

The vertices are at (-3, 0) and (3, 0).

Step-by-step explanation:

For the equation of an ellipse in the form
`\((x^2)/(a^2) + (y^2)/(b^2) = 1\)`, the center is at the origin (0,0), which indicates the ellipse is centered at (0,0). The square root of the denominator of x² (9) gives the length of the major axis (2a), so
`\(2a = 2 * 3 = 6\)`, making a = 3.

Similarly, the square root of the denominator of y² (29) provides the length of the minor axis (2b), so
`\(2b = 2 * √(29)\)`. The vertices are located on the major axis, a distance of 'a' units from the center in both positive and negative x-directions. Therefore, the vertices are at (-3, 0) and (3, 0).