Determine the foci coordinates, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse (x2/49) + (y2/36) = 1

As an experienced tutor registered on UrbanPro, I'm well-versed in helping students tackle challenging mathematical problems. Let's break down this question about ellipses step by step. Given the equation of the ellipse: x249+y236=149x2+36y2=1 First, let's identify the key components of the ellipse: Center: The center of the ellipse is at the origin (0, 0) since there are no constants added or subtracted to the x and y terms. Foci: The foci of the ellipse can be calculated using the formula c=a2−b2c=a2−b2 , where cc is the distance from the center to the focus, aa is the length of the semi-major axis, and bb is the length of the semi-minor axis. In this case, a=7a=7 and b=6b=6, so c=49−36=13c=49−36=13. Therefore, the foci coordinates are (±13,0)(±13 ,0). Vertices: The vertices of the ellipse lie on the major axis. Since the major axis is along the x-axis, the vertices coordinates are (±a,0)(±a,0), where a=7a=7. So the vertices are (7,0)(7,0) and (−7,0)(−7,0). Length of Major Axis: The length of the major axis is 2a2a, which in this case is 2×7=142×7=14. Length of Minor Axis: The length of the minor axis is 2b2b, which in this case is 2×6=122×6=12. Eccentricity (ee): Eccentricity of an ellipse is calculated as e=cae=ac, where cc is the distance from the center to the focus and aa is the length of the semi-major axis. In this case, e=137e=713 . Length of Latus Rectum: The length of the latus rectum is calculated as 2b2/a2b2/a, where aa and bb are the lengths of the semi-major and semi-minor axes respectively. So in this case, it's 2×6272×762. So, to sum up: Foci coordinates: (±13,0)(±13,0) Vertices: (7,0)(7,0) and (−7,0)(−7,0) Length of Major Axis: 14 Length of Minor Axis: 12 Eccentricity: 137713 Length of Latus Rectum: 2×6272×762 Understanding these properties helps in visualizing and understanding the geometry of the ellipse. If you need further clarification or assistance, feel free to ask! And remember, UrbanPro is a fantastic resource for finding experienced tutors like myself for online coaching in various subjects. read less