8.8 Conics With Equations in the Form

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1 8.8 Conics With Equations in the Form ax + b + gx + f + c = 0 The CF-18 Hornet is a supersonic jet flown in Canada. It has a maximum speed of Mach 1.8. The speed of sound is Mach 1. When a plane like the Hornet breaks the sound barrier, it produces a shock wave in the shape of a cone. For a plane fling parallel to the ground, the shock wave, or sonic boom, intersects the ground in a branch of a hperbola. The standard form of the equation of a conic provides a convenient wa to identif the conic and sketch the graph. The equation of a conic can also be written in the form ax + b + gx + f + c = 0. For example, an equation in standard form for a hperbola is (x + ) ( 1) = 1 9 Multipl both sides b 36: (x + ) 9( 1) = 36 Expand and simplif: (x + x + ) 9( + 1) = 36 x + 16x = 36 x x = 0 In general, ax + b + gx + f + c = 0 is a quadratic equation when a and b are not both equal to zero. INVESTIGATE & INQUIRE 1. Cop and complete the table b expanding and simplifing each equation. Write each result in the form ax + b + gx + f + c = 0. a) (x 3) Standard Form ax + b + gx + f + c = 0 ( + ) + =1 5 b) c) d) (x 3) + ( 1) = = (x 1) (x 1) 9 ( + 1) = Conics With Equations in the Form ax + b + gx + f + c = 0 MHR 665

2 . Identif each equation in standard form in question 1 as the equation of a circle, ellipse, hperbola, or parabola. 3. a) For the circle written in the form ax + b + gx + f + c = 0, what do ou notice about the values of a and b? b) For a circle, if f = 0 and g = 0, what are the coordinates of the centre?. a) For the ellipse written in the form ax + b + gx + f + c = 0, what do ou notice about the signs of a and b? b) For an ellipse, if f = 0 and g = 0, what are the coordinates of the centre? 5. a) For the parabola written in the form ax + b + gx + f + c = 0, what do ou notice about the values of a and b? b) What is alwas true for the value of a or b for a parabola? 6. a) For the hperbola written in the form ax + b + gx + f + c = 0, what do ou notice about the signs of a and b? b) For a hperbola, if f = 0 and g = 0, what are the coordinates of centre? 7. If ou are given the equation of a conic in the form ax + b + gx + f + c = 0, how can ou identif the tpe of conic, without rewriting the equation in standard form? 8. Without rewriting in standard form, identif each of the following conics. a) 3x + x + 50 = 0 b) 16x 3x 10 5 = 0 c) x + + 8x + = 0 d) x + 6x + 1 = 0 The tpe of conic represented b an equation in the form ax + b + gx + f + c = 0 can be identified using the signs and values of a and b. For a circle, a = b. For an ellipse, a and b have the same sign, and a b. For a parabola, either a = 0 or b = 0. For a hperbola, a and b have opposite signs. EXAMPLE 1 Sketching the Graph of a Conic a) Identif the tpe of conic whose equation is x x = 0. b) Write the equation in standard form. c) Determine the ke features and sketch the graph. 666 MHR Chapter 8

3 SOLUTION a) Since a = and b = 9, a and b have the same sign, with a b. The conic is an ellipse. b) To write the equation in standard form, complete the square for both variables. x x = 0 Add 11 to both sides: x x + 18 = 11 Group the x and terms: x 16x = 11 Remove common factors: (x x) + 9( + ) = 11 Complete the square: (x x + ) + 9( ) = 11 (x ) ( + 1) 9 = 11 (x ) + 9( + 1) = 36 Divide both sides b 36: (x ) ( + 1) + = 1 9 ( + 1) The equation in standard form is (x ) + = 1. 9 c) The equation is in the form (x h) ( k) + = 1. a b The ellipse is centred at (h, k), or (, 1), and the major axis is parallel to the x-axis. a = 9, so a = 3 b =, so b = The major axis, which is parallel to the x-axis, has a length of a, or 6. The minor axis, which is parallel to the -axis, has a length of b, or. The vertices are V 1 (h a, k) and V (h + a, k). Substitute the values of h, k, and a. The vertices are V 1 ( 3, 1) and V ( + 3, 1), or V 1 ( 1, 1) and V (5, 1). The co-vertices are (h, k b) and (h, k + b). Substitute the values of h, k, and b. The co-vertices are (, 1 ) and (, 1 + ), or (, 3) and (, 1). 8.8 Conics With Equations in the Form ax + b + gx + f + c = 0 MHR 667

4 The foci are F 1 (h c, k) and F (h + c, k). To find c, we use a = b + c, with a = 3 and b =. a = b + c 3 = + c 9 = + c 5 = c 5 = c The coordinates of the foci are ( 5, 1) and ( + 5, 1), or approximatel ( 0., 1) and (., 1). Plot the vertices and co-vertices. Draw a smooth curve through the points. Label the foci and the graph. x x = 0 (, 1) x V 1 ( 1, 1) (, 1) (, 3) F 1 ( 5, 1) V (5, 1) F ( + 5, 1) EXAMPLE Sketching the Graph of a Conic a) Identif the tpe of conic whose equation is + 8x + 15 = 0. b) Write the equation in standard form. c) Determine the ke features and sketch the graph. SOLUTION a) Since a = 0 and b 0, the conic is a parabola. b) To write the equation in standard form, complete the square for the variable. + 8x + 15 = 0 Add 15 to both sides: + 8x + = 15 Group the terms: + + 8x = 15 Complete the square: x = 15 ( + 1) 1 + 8x = 15 ( + 1) + 8x = 16 Rearrange: 8x 16 = ( + 1) Remove a common factor: 8(x ) = ( + 1) Divide both sides b 8: x = 1 8 ( + 1) The equation in standard form is x = 1 8 ( + 1). 668 MHR Chapter 8

5 c) The equation is in the form x h = 1 ( k). p The vertex is V(h, k). h =, k = 1, so the vertex is V(, 1). Find the value of p. From the equation, 1 = 1 p 8 p = ( ) p = p < 0, so the parabola opens left. The focus F(h + p, k) is ( + ( ), 1) or (0, 1). The directrix is x = h p x = ( ) x = The axis of smmetr is = k or = 1. Sketch and label the graph. 6 x = 0 6 x F(0, 1) V(, 1) 6 + 8x + 15 = 0 EXAMPLE 3 Shock Wave A shock wave from an aircraft that breaks the sound barrier intersects the ground in a curve with the equation x + x + 36 = 0. a) Identif the tpe of conic. b) Write the equation in standard form. c) Determine the ke features and sketch the graph. SOLUTION a) Since a and b have opposite signs, the conic is a hperbola. 8.8 Conics With Equations in the Form ax + b + gx + f + c = 0 MHR 669

6 b) To write the equation in standard form, complete the square for both variables. x + x + 36 = 0 Add 36 to both sides: x + x + = 36 Group the x and terms: x + x + = 36 Remove a common factor: x + x ( 6) = 36 Complete the square: x + x + ( ) = 36 (x + ) ( 3) + 36 = 36 (x + ) ( 3) = Divide both sides b : (x + ) ( 3) = 1 The equation in standard form is (x + ) ( 3) = 1. (x h) ( k) c) The equation is in the form = 1. a b The centre is C(h, k) = (, 3). The transverse axis is parallel to the x-axis. a =, so a = b = 1, so b = 1 The vertices are V 1 (h a, k) and V (h + a, k). Substitute the values for h, k, and a. V 1 (, 3) and V ( +, 3) or V 1 (, 3) and V (0, 3). The co-vertices are (h, k b) and (h, k + b). Substitute the values for h, k, and b. The coordinates of the co-vertices are (, 3 1) and (, 3 + 1) or (, ) and (, ). The length of the transverse axis is a = () = The length of the conjugate axis is b = (1) = 670 MHR Chapter 8

7 The coordinates of the foci are F 1 (h c, k) and F (h + c, k). To find c, use c = a + b, with a = and b = 1. c = + 1 = 5 (x + ) c = 5 The coordinates of the foci are ( 5, 3) and ( + 5, 3), or approximatel (., 3) and (0., 3). Sketch and label the graph. F 1 ( 5, 3) V 1 (, 3) ( 3) = 1 8 (, ) (, 3) (, ) F ( + 5, 3) V (0, 3) x Ke Concepts The equation of a conic can be written in the form ax + b + gx + f + c = 0, where a and b are not both equal to zero. The tpe of conic represented b an equation in the form ax + b + gx + f + c = 0 can be identified using the signs and values of a and b. * For a circle, a = b. * For an ellipse, a and b have the same sign, and a b. * For a parabola, a = 0 or b = 0. * For a hperbola, a and b have opposite signs. To graph a conic section whose equation is in the form ax + b + gx + f + c = 0, first use the method of completing the square to write the equation in standard form. Communicate Your Understanding 1. Identif each of the following conics. a) x + 16 = 0 b) x x 6 3 = 0 c) x + x + 8 = 0 d) x 9 1x = 0. Describe how ou would write an equation in standard form for the conic defined b x x = Conics With Equations in the Form ax + b + gx + f + c = 0 MHR 671

8 Practise A 1. Identif the tpe of conic b inspection. a) x 6x + = 0 b) x + 6x 3 = 0 c) x + 5x = 0 d) 3 + 6x 6 9 = 0 e) x 3 6x 1 = 0 f) 3x + 3x = 0 g) x 6x + 9 = 0. For each of the following equations, i) identif the tpe of conic ii) write the equation in standard form iii) determine the ke features and sketch the graph a) x + x 6 15 = 0 b) x + + x = 0 c) x + 6x = 0 d) 8x + 1 = 0 e) x x = 0 f) x + + x 6 3 = 0 g) x + x + 1 = 0 h) + x + 8 = 0 i) x 6x = 0 b) c) d) e) x 8 x x 3. For each conic, write an equation in standard form and in the form ax + b + gx + f + c = 0. a) x x f) 67 MHR Chapter x

9 Appl, Solve, Communicate B. Inquir/Problem Solving a) For the equation x + b = 0, determine the value(s) of b that result in an equation of i) a circle ii) a parabola iii) an ellipse iv) a hperbola b) Give an example to illustrate each answer in part a). 5. a) For the equation ax + 9 = 0, determine the value(s) of a that will result in an equation of i) a circle ii) a parabola iii) an ellipse iv) a hperbola b) Give an example to illustrate each answer in part a). 6. Inquir/Problem Solving The three squares in the diagram are centred at the same point. The red border has the same area as the smallest square. a) How are p and q related? b) Graph the relation. p q cm 7. Application When a plane breaks the sound barrier, a shock wave in the shape of a cone is produced. If the plane is fling parallel to the ground, the shock wave intersects the ground in a branch of a hperbola. a) Use our knowledge of the intersection of a plane and a cone to explain wh the intersection is a branch of a hperbola. b) If the intersection of a shock wave with the ground can be modelled b the equation x + 5 8x = 0, describe how the plane is fling. c) Is it possible for the intersection of a shock wave with the ground to be modelled b the equation x = 0? Explain. 8.8 Conics With Equations in the Form ax + b + gx + f + c = 0 MHR 673

10 C 8. Degenerate conics The rules for identifing a tpe of conic from its equation do not alwas appl. For example, in x + + x + 5 = 0, a = b, so the equation appears to model a circle. Completing the square gives x + + x + 5 = 0 x + x = 0 x + x = 0 (x + 1) + ( ) = 0 The ordered pair ( 1, ) satisfies the equation. The solution is ( 1, ). Therefore, the graph of x + + x + 5 = 0 is a point, not a circle. Because the equation appears to model a circle, the graph is referred to as a degenerate circle. Changing the equation to x + + x + 6 = 0 and completing the square gives (x + 1) + ( ) = 1. This equation has no solution, since the sum of two squares cannot be negative. So, the equation x + + x + 6 = 0 is degenerate. i) Identif the tpe of conic that each of the following equations appears to model. ii) Verif that each equation is degenerate. iii) Graph the equation, if possible. a) x + 8x = 0 b) x + x = 0 c) 3x + 6x + 39 = 0 d) 9x + 18x + 6 = 0 LOGIC Power What are the four moves that X should not pla, if X wants to stop O from winning? 67 MHR Chapter 8