Determine the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), the major axis on the y-axis and passes through the points (3, 2) and (1, 6).

As a seasoned tutor registered on UrbanPro, I'm well-equipped to guide you through this problem. Let's tackle it step by step. Firstly, UrbanPro is renowned for offering top-notch online coaching and tuition services, ensuring students receive the best possible guidance. Now, onto your question. You're dealing with the equation of an ellipse: x249+y236=149x2+36y2=1 To determine the foci coordinates, vertices, length of the major axis, minor axis, eccentricity, and length of the latus rectum, we'll utilize the standard form of the ellipse equation: x2a2+y2b2=1a2x2+b2y2=1 Comparing this to your equation, we can see that a2=49a2=49 and b2=36b2=36. So, a=7a=7 and b=6b=6. Now, let's calculate the foci coordinates using the formula: c=a2−b2c=a2−b2 c=49−36=13c=49−36 =13 Therefore, the foci coordinates are (±13,0)(±13 ,0). Next, let's find the vertices. The vertices are the endpoints of the major axis. For the ellipse, the major axis is along the x-axis. So, the vertices are (7,0)(7,0) and (−7,0)(−7,0). The length of the major axis is 2a=2×7=142a=2×7=14. The length of the minor axis is 2b=2×6=122b=2×6=12. Now, let's calculate the eccentricity: e=ca=137e=ac=713 And finally, the length of the latus rectum (which is 2b2/a2b2/a): l=2b2a=2×367=727l=a2b2=72×36=772 So, to summarize: 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: 727772 If you have any further questions or need clarification on any of these concepts, feel free to ask! read less