The Ellipse - Cartesian and parametric forms

The Ellipse - Cartesian and parametric forms

The Ellipse - Cartesian Form

• The Cartesian form of an ellipse is given by the equation x²/a² + y²/b² = 1, where a and b are the lengths of the semi-major and semi-minor axes, respectively.
• If the center of the ellipse is not at the origin but at the coordinates (h,k), the equation becomes (x-h)²/a² + (y-k)²/b² = 1.
• a is always greater than or equal to b. If a equals b, the ellipse is a circle.
• The points (a, 0) and also (-a, 0) on the ellipse are known as the vertices.

The Ellipse - Parametric Form

• The parametric form of an ellipse is given by x = a cos θ, y = b sin θ, where θ is the parameter, also known as the eccentric angle.
• θ is an angle measured in the positive direction from the positive x-axis to the line segment connecting the center of the ellipse to a point P(x,y) on the ellipse.
• A brilliant property of the parametric form is that it traces the whole ellipse as θ varies from 0 to 2π.
• The parametric form is handy for solving geometry problems involving tangents, areas, and arc lengths in the context of an ellipse.

Both the Cartesian and Parametric forms are crucial for understanding the algebraic and geometric properties of an ellipse. Understanding the relationship between these two forms is essential to interpret an ellipse in the Further Pure 1 module.