Ellipse Equations and Properties

Equation of Ellipse Forms

• Standard Form: ( x^2 + \frac{y^2}{25} = 1 ) represents a vertical ellipse with semi-minor axis 5.
• Standard Form: ( \frac{x^2}{25} + y^2 = 1 ) represents a horizontal ellipse with semi-major axis 5.
• Center Offset: ( \frac{(x-2)^2}{9} + y^2 = 1 ) describes a horizontal ellipse centered at (2, 0) with semi-major axis 3.
• General Center Shift: ( \frac{(x-3)^2}{4} + \frac{(y+7)^2}{9} = 1 ) indicates an ellipse centered at (3, -7) with axes 2 and 3.
• Rescaled Ellipse: ( \frac{49x^2}{25} + \frac{y^2}{9} = 1 ) shows transformation into standard form emphasizing the shaping of axes.

General Form of Ellipses

• Rotated Ellipses: Terms like ( x^2 + xy + y^2 = 1 ) imply rotation; the criterion for predicting an ellipse is ( B^2 - 4AC < 0 ).
• Rotation with Additional Terms: ( x^2 + xy + y^2 + 4x = 1 ) contains both rotation indicator and linear components affecting the position and orientation.

Properties and Characteristics

• Countered Rotated Forms: Expressions like ( x^2 - xy + y^2 + 2x + 2y = 1 ) remain rotational; ( B^2 - 4AC < 0 ) confirms the presence of an ellipse despite added terms.
• Visual Representation: Adjustments in coefficients affect the width and height of the ellipse visually represented on a graph.

Area of an Ellipse

• Area Calculation: The area formula of an ellipse is given as ( \pi ab ), where ( a ) and ( b ) are the semi-major and semi-minor axes respectively.

Inverse Relationships

• Inverse Notation: Recognize that presentations like ( \frac{16x^2}{9} + \frac{36y^2}{25} = 1 ) can be rephrased using reciprocal relationships in denominators.

Key Takeaway

• Quadratic Equation: The general form ( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 ) is vital for identifying conic types and characteristics, with attention to discriminant criteria for distinguishing ellipses from other conics.

This quiz focuses on the equations of ellipses, including their standard and general forms. It covers various aspects such as center offsets, rotations, and properties associated with ellipses. Challenge your understanding of how these equations represent different types of ellipses and their characteristics.