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Chapter 6: Analytic Geometry. 6.1 Circles and Parabolas 6.2 Ellipses and Hyperbolas 6.3 Summary of the Conic Sections 6.4 Parametric Equations. 6.3 Summary of the Conic Sections. Conic sections presented in this chapter are of the form where either A or C must be nonzero.
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Chapter 6: Analytic Geometry 6.1 Circles and Parabolas 6.2 Ellipses and Hyperbolas 6.3 Summary of the Conic Sections 6.4 Parametric Equations
6.3 Summary of the Conic Sections • Conic sections presented in this chapter are of the form where either A or C must be nonzero.
6.3 Summary of the Conic Sections Last 3 rows of the Table on pg 6-52
6.3 Determining Type of Conic Section • To recognize the type of conic section, we may need to transform the equation. Example Decide on the type of conic section represented by each equation.
6.3 Determining Type of Conic Section Solution (a) This equation represents a hyperbola centered at the origin.
6.3 Determining Type of Conic Section • Coefficients of the x2- and y2-terms are unequal and both positive. This equation may be an ellipse. Complete the square on x and y. This is an ellipse centered at (2, –3).
6.3 Determining Type of Conic Section • Coefficients of the x2- and y2-terms are both 1. This equation may be a circle. Complete the square on x and y. This equation is a circle with radius 0; that is, the point (4, –5).
6.3 Determining Type of Conic Section • Since only one variable is squared, x2, the equation represents a parabola. Solve for y (the variable that is not squared) and complete the square on x (the squared variable). The parabola has vertex (3, 2) and opens downward.
6.3 Eccentricity A conic is the set of all points P(x, y) in a plane such that the ratio of the distance from P to a fixed point and the distance from P to a fixed line is constant. • The constant ratio is called the eccentricity of the conic, written e.
6.3 Eccentricity of Ellipses and Hyperbolas • Ellipses and hyperbolas have eccentricity where c is the distance from the center to a focus. • For ellipses, a2 > b2 and • Note that ellipses with eccentricity close to 0 have a circular shape because b a asc 0.
6.3 Finding Eccentricity of an Ellipse Example Find the eccentricity of Solution Since 16 > 9, let a2 = 16, giving a = 4.
6.3 Figures Comparing Different Eccentricities of Ellipses and Hyperbolas
6.3 Finding Equations of Conics using Eccentricity Example Find an equation for each conic with center at the origin. • Focus at (3, 0) and eccentricity 2 • Vertex at (0, –8) and e = Solution (a)e = 2 > 1, this conic is a hyperbola with c = 3.
6.3 Finding Equations of Conics using Eccentricity The focus is on the x-axis, so the x2-term is positive. The equation is
6.3 Finding Equations of Conics using Eccentricity • Since the conic is an ellipse. The vertex at (0, –8) indicates that the vertices lie on the y-axis and a = 8. Since the equation is
6.3 Applying an Ellipse to the Orbit of a Planet Example The orbit of the planet Mars is an ellipse with the sun at one focus. The eccentricity of the ellipse is 0.0935, and the closest Mars comes to the sun is 128.5 million miles. Find the maximum distance of Mars from the sun. Solution Using the given figure, Mars is - closest to the sun on the right - farthest from the sun on the left Therefore, - smallest distance is a – c - greatest distance is a + c
6.3 Applying an Ellipse to the Orbit of a Planet Since a – c = 128.5, c = a – 128.5. Using e = 0.0935, we find a. The maximum distance of Mars from the sun is about 155.1 million miles.
