Conic Sections and Polar Graphing Lab Part 1 - Circles
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1 MAC 1114 Name Conic Sections and Polar Graphing Lab Part 1 - Circles 1. What is the standard equation for a circle with center at the origin and a radius of k? 3. Consider the circle x + y = 9. a. What is the circle's center? b. What is the circle's radius? c. Convert the rectangular equation for the circle into a polar equation, showing your work in the space below and then graph the circle in polar the circle to fill a significant portion of the window. 4. Consider the circle x 3 + y + = 4. a. What is the circle's center? b. What is the circle's radius? c. Convert the rectangular equation for the circle into a polar equation, showing your work in the space below and then graph the circle in polar the circle to fill a significant portion of the window. 5. Give both the rectangular and polar equations for the circle at right. a. What is bxy, g when θ = 3?b. What is r when θ = k? Steve Boast
2 Part - Ellipses 1. The equation for an ellipse whose center is at the origin is + = 1. The a b constants a and b represent the lengths of the semi-major and semi-minor axes of the ellipse. a. If a>b, then the ellipse is elongated along the -axis, the length of the major axis is, and the length of the minor axis is. Note: c = a - b b. If b>a, then the ellipse is elongated along the -axis, the length of the major axis is, and the length of the minor axis is. 3. Consider the ellipse + = a. What is the ellipse's center? b. Major axis = and minor axis =. c. Convert the rectangular equation for the ellipse into a polar equation, showing your work in the space below and then graph the ellipse in polar the ellipse to fill a significant portion of the window. x + 4 y Consider the ellipse + = a. What is the ellipse's center? b. Major axis = and minor axis =. c. Convert the rectangular equation for the ellipse into a polar equation, showing your work in the space below and then graph the ellipse in polar the ellipse to fill a significant portion of the window. Steve Boast Page
3 Part - Ellipses (cont.) 5. Give both the rectangular and polar equations for the ellipse at right. a. What is r when θ =? b. What is r when θ = 5? 4 6 c. What is ( xy, ) when θ =? Part 3 - Hyperbolas 1. The equation for a hyperbola whose center is at the origin is = 1. The constants a and b represent the lengths of the a b semi-major and semi-minor axes of the hyperbola. a. If a>b, then the hyperbola opens along the -axis, the length of the major axis is, and the length of the minor axis is. Note: c = a + b b. If b>a, then the hyperbola opens along the -axis, the length of the major axis is, and the length of the minor axis is. 3. Consider the hyperbola 9x 16y = 144. Write the equation in standard form. a. What is the hyperbola's center? b. Major axis = and minor axis =. Steve Boast Page 3
4 Part 3 - Hyperbolas (cont.) c. Convert the rectangular equation for the hyperbola into a polar equation, showing your work in the space below and then graph the hyperbola in polar mode in the space at right. Please find and indicate a window that allows the hyperbola to fill a significant portion of the window. x 4 y Consider the hyperbola = 16. (Change to standard form first.) 1 4 a. What is the hyperbola's center? b. Major axis = and minor axis =. c. Find the polar equation for the hyperbola. a. What is r when θ =? b. What is ( xy, ) when θ =? 6 5. Write the following hyperbola in standard form: 4x 9y = 1 a. What is the center of the hyperbola? b. What is the length of the semi-major and semi-minor axis? 6. Write the following hyperbola in standard form: 3 y 3 8 x + 7 = 18 a. What is the center of the hyperbola? b. What is the length of the semi-major and semi-minor axis? Steve Boast Page 4
5 The Four Conic Sections Color in the four conic sections in the figure below using different colors for each conic section. 1. Name the four conic sections: a. b c. d.. Why do you suppose they are called conic sections? 3. What is the relationship between the plane determing each conic section and the two cones? a. Parabola - b. Ellipse - c. Hyperbola - d. Circle - Steve Boast Page 5
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