Final answer:
To find the vertices of the given ellipse equation, we identify the center (-9, -8) and use the lengths of the semi-major (√46) and semi-minor axes (√9) to find the vertices along the x-axis ((-9 ± √46, -8)) and y-axis ((-9, -8 ± √9)).
Step-by-step explanation:
The student is asking for the vertices of the ellipse given by the equation ((x+9)²)/46 + ((y+8)²)/9 = 1. To find the vertices, we first identify the center of the ellipse, which is at the point (-9, -8) based on the standard form of an ellipse equation. The next step is to determine the lengths of the semi-major and semi-minor axes from the denominators of the equation. In this case, the semi-major axis is √46 and the semi-minor axis is √9.
The vertices on the major axis lie at a distance of the semi-major axis from the center, along the direction of the x-axis in this case. Thus, the vertices are (-9 ± √46, -8). Simplifying further, we get the vertices as approximately (-2.22, -8) and (-15.78, -8).
The vertices on the minor axis lie at a distance of the semi-minor axis from the center, along the direction of the y-axis. They are located at (-9, -8 ± √9), which simplifies to (-9, -5) and (-9, -11).