Conic Sections Flashcards

Conic Sections Overview

• Conic sections are the curves obtained by intersecting a cone with a plane. The main types are circles, ellipses, hyperbolas, and parabolas.

Circle

• Defined by the equation: ( x² + y² = 16 ).
• Represents a circle with a radius of 4, centered at the origin (0,0).

Ellipse

• Classified into vertical and horizontal types based on standard equations.
• Vertical ellipse equation example: ( (x-5)² / 25 + (y-3)² / 16 = 1 ).

Hyperbola

• Two types: horizontal and vertical, identified by their equations.
• Example of a vertical hyperbola: ( (y-4)² / 16 - (x+4)² / 9 = 1 ).

Parabola

• Parabolic equations can indicate their direction (up, down, left, right).
• Example: ( x² = 16(y-5) ) represents an upward-opening parabola.

Direction of Parabolas

• Downward direction is indicated by ( y-2 = -8(x+3)² ).
• Leftward opening is indicated by ( (y-2)² = -8x ).
• Upward direction found in ( x² + 6x + 8y + 2 = 0 ).
• Rightward opening noted in ( (4x+8) / 4 = (3y+6)² / 9 ).

Types of Ellipses

• Vertical Ellipse example: ( (x² / 25) + (y² / 49) = 0 ).
• Horizontal Ellipse example: ( (y-4)² / 36 + (x² + 10)² / 49 = 1 ).

Types of Hyperbolas

• Horizontal Hyperbola example: ( x² - 4y² - 4x - 8y - 16 = 0 ).
• Vertical Hyperbola example: ( (y-4)² / 16 - (x+4)² / 9 = 1 ).

Centers of Conic Sections

• Center of a vertical ellipse ( (x² / 25) + (y² / 49) = 0 ) is at (0, 0).
• Center of a horizontal hyperbola ( x² - 4y² - 4x - 8y - 16 = 0 ) is at (2, -1).
• Center of a parabola given by ( x² + 6x + 8y + 1 = 0 ) is at (-3, 1).
• Another center found in ( (x²-3)/ 25 + (y²+1) / 49 = 0 ) is at (3, -1).
• Center from ( (y-2)² = 28x ) is at (0, 2).

Axes Lengths

• Length of major axis from ( (y-4)² / 36 + (x² + 10)² / 49 = 1 ) is 14.
• Major axis length of another ellipse ( (y-4)² / 64 + (x² + 10)² / 144 = 1 ) is 24.
• Length of minor axis from ( (x-4)² / 16 + (y² + 10)² / 25 = 1 ) is 8.
• Length of semi-minor axis from ( (x-4)² / 36 + (y² + 10)² / 49 = 1 ) is 6.