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Chapter Analysis
Intermediate28 pages • EnglishQuick Summary
The chapter 'Conic Sections' introduces various curves that can be derived from the intersection of a plane with a double napped right circular cone. It covers the definitions and properties of circles, ellipses, parabolas, and hyperbolas along with their standard equations and geometrical significance. The chapter emphasizes the application of these curves in real-world contexts and provides exercises to understand their properties and equations.
Key Topics
- •Definition of conic sections
- •Standard equations of circles, ellipses, parabolas, and hyperbolas
- •Geometric properties and applications of conic sections
- •Eccentricity and its implications
- •Derivation of conic section equations
- •Degenerated conic sections
Learning Objectives
- ✓Understand the concept and formation of conic sections
- ✓Identify and apply the standard equations of conic sections
- ✓Analyze the geometric properties of different conic sections
- ✓Solve mathematical problems involving the properties and equations of conic sections
- ✓Interpret the applications of conic sections in real-life scenarios
Questions in Chapter
In each of the Exercises 1 to 6, find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.
Page 202
In each of the Exercises 7 to 15, find the equations of the hyperbola satisfying the given conditions.
Page 202
Does the point (–2.5, 3.5) lie inside, outside or on the circle x2 + y2 = 25?
Page 182
Additional Practice Questions
What is the standard equation of a circle with center at the origin and radius r?
easyAnswer: The standard equation is x2 + y2 = r2.
Derive the equation of an ellipse with the center at the origin, major axis along the x-axis, and semi-major axis length 'a' and semi-minor axis length 'b'.
mediumAnswer: The equation of the ellipse is x^2/a^2 + y^2/b^2 = 1.
Explain the significance of the eccentricity of conic sections.
mediumAnswer: The eccentricity of a conic section determines its shape. For an ellipse, 0 < e < 1; for a parabola, e = 1; and for a hyperbola, e > 1.
Find the length of the latus rectum of a hyperbola given by x^2/a^2 - y^2/b^2 = 1.
hardAnswer: The length of the latus rectum of the hyperbola is 2b^2/a.
What happens to a conic section when the intersecting plane passes through the vertex of the cone?
hardAnswer: When the plane passes through the vertex, it results in degenerated conic sections: a point, a line, or intersecting lines.