Questions and Answers
Identify the conic section in the equation: $x² + y² = 16$.
Identify the conic section in the equation: $(x-5)² / 25 + (y-3)² / 16 = 1$.
Identify the conic section in the equation: $(y-1)² / 9 - (x-2)² / 16 = 1$.
Identify the conic section in the equation: $x² = 16(y-5)$.
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Find the direction of the parabola in the equation: $y-2 = -8(x+3)²$.
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Find the direction of the parabola in the equation: $(y-2)² = -8x$.
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Find the direction of the parabola in the equation: $x² + 6x + 8y + 2 = 0$.
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Find the direction of the parabola in the equation: $(4x + 8) / 4 = (3y + 6)² / 9 = 0$.
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Find what kind of ellipse is in the equation: $(x² / 25) + (y² / 49) = 0$.
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Find what kind of ellipse is in the equation: $(y-4)² / 36 + (x² + 10)² / 49 = 1$.
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Find what kind of hyperbola is in the equation: $x² - 4y² - 4x - 8y - 16 = 0$.
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Find what kind of hyperbola is in the equation: $(y-4)² / 16 - (x+4)² / 9 = 1$.
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Find the ordered pair that is the center of the equation: $(x² / 25) + (y² / 49) = 0$.
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Find the ordered pair that is the center of the equation: $x² - 4y² - 4x - 8y - 16 = 0$.
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Find the ordered pair that is the center of the equation: $x² + 6x + 8y + 1 = 0$.
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Find the ordered pair that is the center of the equation: $(x² - 3) / 25 + (y² + 1) / 49 = 0$.
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Find the ordered pair that is the center of the equation: $(y-2)² = 28x$.
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Find the length of the major axis in the equation: $(y-4)² / 36 + (x² + 10)² / 49 = 1$.
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Find the length of the major axis in the equation: $(y-4)² / 64 + (x² + 10)² / 144 = 1$.
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Find the length of the minor axis in the equation: $(x-4)² / 16 + (y² + 10)² / 25 = 1$.
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Find the length of the semi-minor axis in the equation: $(x-4)² / 36 + (y² + 10)² / 49 = 1$.
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Study Notes
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.
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Test your knowledge of conic sections with these flashcards. Each card presents a conic section and its corresponding equation for identification. Challenge yourself to recognize circles, ellipses, hyperbolas, and parabolas.
