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Questions and Answers
What is the value of eccentricity for a parabola?
What is the value of eccentricity for a parabola?
What is the characteristic that distinguishes a circle from other conic sections?
What is the characteristic that distinguishes a circle from other conic sections?
What is the equation of a parabola in standard form?
What is the equation of a parabola in standard form?
What is the term for the line that passes through the vertex and is perpendicular to the directrix?
What is the term for the line that passes through the vertex and is perpendicular to the directrix?
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What is the characteristic shared by ellipses and hyperbolas?
What is the characteristic shared by ellipses and hyperbolas?
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What is the angle of intersection between the plane and the cone's axis that forms a parabola?
What is the angle of intersection between the plane and the cone's axis that forms a parabola?
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Study Notes
Conic Sections
Definition
- A conic section is a curve obtained by intersecting a cone with a plane
- The resulting curve depends on the angle of intersection between the plane and the cone
Types of Conic Sections
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Circle: formed when the plane is perpendicular to the cone's axis
- Equation: x^2 + y^2 = r^2 (standard form)
- Characteristics: constant radius, no vertex, no focal points
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Ellipse: formed when the plane is inclined at an angle less than 90° to the cone's axis
- Equation: (x^2/a^2) + (y^2/b^2) = 1 (standard form)
- Characteristics: two foci, major and minor axes, eccentricity (e)
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Parabola: formed when the plane is parallel to the cone's generator
- Equation: y = ax^2 + bx + c (standard form)
- Characteristics: one focus, vertex, axis of symmetry
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Hyperbola: formed when the plane is inclined at an angle greater than 90° to the cone's axis
- Equation: (x^2/a^2) - (y^2/b^2) = 1 (standard form)
- Characteristics: two foci, transverse and conjugate axes, eccentricity (e)
Key Concepts
- Focus: a point on the conic section that is equidistant from every point on the curve
- Vertex: the lowest or highest point on the conic section (depending on orientation)
- Axis of symmetry: the line that passes through the vertex and is perpendicular to the directrix
- Directrix: a line that is perpendicular to the axis of symmetry and is used to define the conic section
- Eccentricity (e): a measure of how "stretched" the conic section is (e = 0 for a circle, e > 0 for an ellipse, e = 1 for a parabola, e > 1 for a hyperbola)
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Description
Learn about the definition, types, and key concepts of conic sections, including circles, ellipses, parabolas, and hyperbolas. Understand the equations and characteristics of each type of conic section.
