Determine the equation of the ellipse of eccentricity e = 7/9, center C(0;0) and passes through P(8;9/2)

To determine the equation of the ellipse, we need to find the major and minor axes lengths, as well as the position of the foci. Given that the center of the ellipse is at C(0, 0) and it passes through point P(8, 9/2), we can calculate the distance from the center to P using the distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²) d = √((8 - 0)² + (9/2 - 0)²) d = √(64 + 81/4) d = √(256/4 + 81/4) d = √(337/4) Now, we know that the sum of the distances from any point on the ellipse to the foci is equal to the length of the major axis. For an ellipse with eccentricity e, the distance from the center to each focus is a * e, where a is the length of the semi-major axis. In our case, the eccentricity e = 7/9. Let's denote the length of the semi-major axis as a. The distance from the center to one focus is a * e = a * (7/9). Since the distance from the center to P is d = √(337/4), we have the following equation: 2 * a * (7/9) = √(337/4) a = (√(337/4)) * (9/14) Simplifying: a = (3/2) * (√(337/4)). Now, we can use the equation of an ellipse centered at the origin to write the equation in standard form: (x² / a²) + (y² / b²) = 1, where b is the length of the semi-minor axis. Since the center is at C(0, 0), the equation becomes: (x² / a²) + (y² / b²) = 1. Substituting the values of a and b, we have: (x² / [(3/2) * (√(337/4))²]) + (y² / b²) = 1. Simplifying further: (x² / [(3/2) * (337/4)]) + (y² / b²) = 1. Multiplying through by [(3/2) * (337/4)], we get: (x² * (4/337)) + (y² / b²) = 1. Since the eccentricity e = 7/9, we know that b = a * √(1 - e²). Substituting the values of a and e, we have: b = [(3/2) * (√(337/4))] * √(1 - (7/9)²). Simplifying further: b = [(3/2) * (√(337/4))] * √(1 - 49/81). b = [(3/2) * (√(337/4))] * (√(81/81) - √(49/81)). b = [(3/2) * (√(337/4))] * (√(32/81)). b = [(3/2) * (√(337/4))] * (4/9). b = [(3/2) * (√(337/4))] * (4/9). b = (6/9) * (√(337/4)). b = (2/3) * (√(337/4)). Finally, the equation of the ellipse is: (x² * (4/337)) + (y² / [(2/3) * (√(337/4))]²) = 1. Simplifying further: (3/337) * x² + (9/4) * y² = 1. Therefore, the equation of the ellipse with eccentricity e = 7/9, center C(0, 0), and passing through P(8, 9/2) is: (3/337) * x² + (9/4) * y² = 1.