REVIEW OF KEY CONCEPTS

Transcription

1 REVIEW OF KEY CONCEPTS Equations of Loci Refer to the Key Concepts on page Sketch the locus of points in the plane that are cm from a circle of radius 5 cm.. a) How are the lines y = x 3 and y = x 9 related? b) Graph the two lines. c) Graph the locus of points that are equidistant from the two lines. d) Write an equation to represent this locus. 3. Find an equation of the locus of points equidistant from each pair of points. a) (, 1) and ( 3, 5) b) ( 3, 4) and (6, 1) 4. Determine an equation of the locus of points equidistant from the graphs of y = x + and y = x The Circle Refer to the Key Concepts on page Sketch and label each circle. Then, state the domain and range. a) + = 81 b) + = 40 c) (x 3) + (y + 5) = 5 d) (x + 1) + (y 4) = Write an equation in standard form for each circle. a) centre (4, ) and radius 3 b) centre (, 4) and passing through the point (3, 0) c) endpoints of a diameter (1, 4) and (7, ) 7. Earthquake The epicentre of an earthquake is found to be 50 km from a seismic recording station. The direction of the epicentre cannot be determined from a reading at one station. Write an equation to model all possible locations of the earthquake epicentre if the recording station is located at (, 3). Review of Key Concepts MHR 689

2 8.5 The Ellipse Refer to the Key Concepts on page Use the locus definition of the ellipse to write an equation in standard form for each ellipse. a) foci (, 0) and (, 0), sum of focal radii 6 b) foci (0, 3) and (0, 3), sum of focal radii 8 9. Sketch the graph of each ellipse. Label the coordinates of the centre, the vertices, the co-vertices, and the foci. State the domain and range. a) + = b) + = (x 3) ( y + 1) c) + = (x + 1) ( y ) d) + = Write an equation in standard form for each ellipse. a) centre (0, 0), major axis along x-axis, length of major axis 10, length of minor axis 5 b) centre (0, 0), one vertex is (0, 7), one focus is (0, 5) c) foci at (, ) and (, 7), with sum of focal radii 0 d) centre (, 1) and passing through ( 3, 1), (7, 1), (, 1) and (, 4) 11. Mercury Planets orbit the sun in elliptical paths with the sun at one focus. The least distance from the sun to Mercury is 4.60 million kilometres, and the greatest distance from the sun to Mercury is 6.98 million kilometres. Write an equation of the ellipse that models Mercury s orbit about the sun. Assume that the sun is on the x-axis. 8.6 The Hyperbola Refer to the Key Concepts on page Use the locus definition of the hyperbola to find an equation of each hyperbola. a) foci ( 5, 0) and (5, 0), and F 1 P F P = 4 b) foci (0, 4) and (0, 4), and F 1 P F P = 690 MHR Chapter 8

3 13. Sketch the graph of each hyperbola. Label the coordinates of the centre, the vertices, the co-vertices, and the foci. State the domain and range. a) = 1 b) = (x + ) ( y 1) ( y 3) c) = 1 (x 1) d) = Determine an equation in standard form for each hyperbola. a) vertices ( 4, 0), (4, 0), co-vertices (0, 3), (0, 3) b) vertices ( 5, ), ( 5, 6), foci ( 5, 4), ( 5, 8) c) centre (, 1), one focus ( 4, 1), length of conjugate axis 4 d) foci (0, 6) and (0, 6), with constant difference between focal radii Roof arches Hyperbolic arches anchored to the ground support the roof of a sports complex. These arches span a distance of 60 m and have a maximum height of 0 m. a) Find a possible equation for a hyperbola to model one of these arches. b) What is the height of the arch at a horizontal distance of 5 m from the maximum? 8.7 The Parabola Refer to the Key Concepts on page Write an equation in the standard form for the parabola with the given focus and directrix. a) focus (0, 3), directrix y = 3 b) focus ( 3, ), directrix x = 1 c) focus (5, 1), directrix y = 1 d) focus (1, ), directrix y = 4 e) focus (3, ), directrix x = For each of the following parabolas, determine the coordinates of the vertex and the focus, and the equation of the directrix. Sketch the graph and determine the domain and range. a) y = 1 8 b) x = 1 1 c) x 3 = 1 (y + 1) d) y + 1 = 1 4 (x 5) Review of Key Concepts MHR 691

4 18. Use the locus definition of the parabola to write an equation for each parabola. a) focus (4, ), directrix x = b) focus (, 3), directrix y = 1 c) focus (4, 1), directrix x = Satellite dish The focus of a parabolic satellite dish is 5 cm from the vertex. a) Determine an equation in standard form to represent the dish, if the parabola opens up, the vertex is at the origin, and the focus is on the y-axis. b) Determine the width of the dish 10 cm from the vertex. Round the width to the nearest tenth of a centimetre. 0. Football A football player kicks the ball during a field goal attempt. The ball reaches a maximum height of 10 m and travels a horizontal distance of 50 m. Write an equation in standard form to model the parabolic path of the football. 8.8 Conics With Equations in the Form a + b + gx + fy + c = 0 Refer to the Key Concepts on page For each of the following equations, i) identify the conic ii) write the equation in standard form iii) determine the key features and sketch the graph a) + + 6x 4y + 1 = 0 b) x 100y + 4 = 0 c) y 1x + 8 = 0 d) x + 8y + 11 = 0 e) + 6x 10y + 3 = 0 f) 9 36x 8y 16 = 0 g) x + 4y + 7 = 0 h) 4 + 6x + 16y 3 = 0. For each conic, write an equation in standard form and in the form a + b + gx + fy + c = 0. a) y b) y x x 69 MHR Chapter 8

5 c) y d) x y x 8.9 Intersections of Lines and Conics Refer to the Key Concepts on page Solve each system of equations. Round answers to the nearest tenth, if necessary. a) + = 5 b) y = 3 c) + 9 = 81 x + y = 5 x + y = 3 x y = 6 d) = 16 e) + = 64 f) 4 + = 100 x + y = 6 x y = x y = 5 g) y = 4 h) + = 5 i) 9 = 36 4x y = 1 1 x y = 7 3x + y = 1 4. Air-traffic control The radar signals from an air traffic control centre at a small airport can be modelled by the equation + = 30 on a grid with 1 unit equal to 1 km. A plane is flying along a path represented by the equation y = 1 x + 3. Find the length of the plane s path that the control centre can monitor, to the nearest tenth of a kilometre. Review of Key Concepts MHR 693