SUMMARY
The equation of the ellipse with a semi-minor axis \( b = 3 \) and a point \( M(-2\sqrt{5}, 2) \) on it is derived using the standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). The calculation shows that \( a = 2 \), leading to the equation \( \frac{x^2}{4} + \frac{y^2}{9} = 1 \). However, the correct form of the equation, as per the textbook, is \( 9x^2 + 36y^2 = 324 \). The error in the initial calculation stemmed from not multiplying the constant term correctly, which should have been \( 180 + 4a^2 = 9a^2 \).
PREREQUISITES
- Understanding of conic sections, specifically ellipses
- Familiarity with the standard form of an ellipse equation
- Basic algebraic manipulation skills
- Knowledge of coordinate geometry
NEXT STEPS
- Review the derivation of the standard form of an ellipse equation
- Practice solving for parameters \( a \) and \( b \) in ellipse equations
- Learn about transformations of conic sections
- Explore the implications of points on conic sections in coordinate geometry
USEFUL FOR
Students studying conic sections, mathematics educators, and anyone preparing for algebra or geometry examinations.