Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
C Review 10.5-10.7 Conic Sections H General Form of a Conic Equation We usually see conic equations written in General, or Implicit Form: Ax Bxy Cy Dx Ey F 0 2 2 where A, B, C, D, E and F are integers and A, B and C are NOT ALL equal to zero. Please Note: A conic equation written in General Form doesn’t have to have all SIX terms! Several of the coefficients A, B, C, D, E and F can equal zero, as long as A, B and C don’t ALL equal zero. If A, B and C all equal zero, what kind of equation do you have? ... T H I N K... Dx Ey F 0 Linear! So, it’s a conic equation if... • the highest degree (power) of x and/or y is 2 (at least ONE has to be squared) • the other terms are either linear, constant, or the product of x and y • there are no variable terms with rational exponents (i.e. no radical expressions) or terms with negative exponents (i.e. no rational expressions) The values of the coefficients in the conic equation determine the TYPE of conic. Ax Bxy Cy Dx Ey F 0 2 2 What values form an Ellipse? What values form a Hyperbola? What values form a Parabola? Ellipses... Ax Cy Dx Ey F 0 2 2 where A & C have the SAME SIGN NOTE: There is no Bxy term, and D, E & F may equal zero! For example: 2x y 8x 0 2 2 x 2x 3y 6y 0 2 2 2x 2y 8x 6 0 2 2 The General Form of the equations can be converted to Standard Form by completing the square and dividing so that the constant = 1. Ellipses... 2x y 8x 0 2 2 2x y 8x 0 2 2 This is an ellipse since x & y are both squared, and both quadratic terms have the same sign! Center (-2, 0) 2(x 4x 4) y 8 2 2 2(x 2) y 8 2 2 2(x 2)2 y 2 8 8 8 8 (x 2) y 1 4 8 2 2 Hor. Axis = 2 Vert. Axis = √8 Ellipses... In this example, x2 and y2 are both negative (still the same sign), you can see in the final step that when we divide by negative 4 all of the terms are positive. x 3y 2x 6y 0 2 2 x 2 2x 3y 2 6y 0 Vert. axis = 2/√3 (x 2x 1) 3(y 2y 1) 1 3 2 2 (x 1) 3(y 1) 4 2 2 (x 1)2 3(y 1)2 1 4 4 (x 1) (y 1) 1 4 4 3 2 center (-1, 1) 2 Hor. axis = 2 Ellipses…a special case! When A & C are the same value as well as the same sign, the ellipse is the same length in all directions … it is a ... Circle! Radius = √5 2x 2y 8x 6 0 2 2 2(x 2 4x 2) 2y 2 6 4 2(x 2) 2 2y 2 10 (x 2)2 y 2 1 5 5 Center (2, 0) Hyperbola... Ax Cy Dx Ey F 0 2 2 where A & C have DIFFERENT signs. NOTE: There is no Bxy term, and D, E & F may equal zero! For example: 9x 4y 36x 8y 4 0 2 2 x y 6y 5 0 2 2 x 10x 4y 8y 5 0 2 2 Hyperbola... The General, or Implicit, Form of the equations can be converted to Graphing Form by completing the square and dividing so that the constant = 1. 9x 2 4 y 2 36x 8y 4 0 9x 2 4y2 36x 8y 4 0 This is a hyperbola since x & y are both squared, and the quadratic terms have different signs! 9(x 2 4x) 4(y2 2y) 4 9(x 2 4x 4) 4(y 2 2y 1) 4 36 4 9(x 2)2 4(y 1)2 36 9(x 2) 4(y 1) 36 36 36 36 (x 2)2 (y 1)2 1 4 9 2 2 x-axis=2 y-axis=3 Center (2,-1) Hyperbola... In this example, the signs change, but since the quadratic terms still have different signs, it is still a hyperbola! x y 6y 5 0 2 2 x 2 y 2 6y 5 0 x 2 (y 2 6y) 5 x (y 6y 9) 5 9 2 2 x 2 (y 3)2 4 x 2 (y 3)2 4 4 4 4 (y 3)2 x 2 1 4 4 x-axis=2 Center (0,3) y-axis=2 Parabola... A Parabola can be oriented 2 different ways: A parabola is vertical if the equation has an x squared term AND a linear y term; it may or may not have a linear x term & constant: Ax Dx Ey F 0 2 A parabola is horizontal if the equation has a y squared term AND a linear x term; it may or may not have a linear y term & constant: Cy Dx Ey F 0 2 Parabola …Vertical The following equations all represent vertical parabolas in general form; they all have a squared x term and a linear y term: x 4x y 7 0 2 4x 8x y 0 2 x y70 2 x y0 2 Parabola …Vertical To write the equations in Standard Form, complete the square for the x-terms. There are 2 popular conventions for writing parabolas in Graphing Form, both are given below: 0 x 2 4x y 7 0 (x 2 4x 4) y 7 4 0 (x 2)2 y 3 y (x 2)2 3 or y 3 (x 2)2 Vertex (2,3) Parabola …Vertical In this example, the signs must be changed at the end so that the y-term is positive, notice that the negative coefficient of the x squared term makes the parabola open downward. 0 4x 8x y 2 0 4(x 2x 1) y 4 2 0 4(x 1) y 4 2 y 4(x 1) 4 2 y 4(x 1)2 4 or y 4 4(x 1)2 Vertex (-1,4) Parabola …Horizontal The following equations all represent horizontal parabolas in general form, they all have a squared y term and a linear x term: y 8y 2x 18 0 2 x y 3 0 2 3y 6y x 2 0 2 y x0 2 Parabola …Horizontal To write the equations in Standard Form, complete the square for the y-terms. There are 2 popular conventions for writing parabolas in Standard Form, both are given below: 0 y 2 8y 2x 18 0 (y 2 8y 16) 2x 18 16 0 (y 4) 2x 2 2 2x (y 4) 2 2 1 x (y 4)2 1 2 or 0 (y 4) 2 2(x 1) 2(x 1) (y 4) 2 Vertex (1,-4) Parabola …Horizontal In this example, the signs must be changed at the end so that the x-term is positive; notice that the negative coefficient of the y squared term makes the parabola open to the left. 0 x y2 3 0 y2 x 3 x y2 3 x y2 3 or (x 3) y 2 Vertex (3,0) What About the term Bxy? Ax Bxy Cy Dx Ey F 0 2 2 None of the conic equations we have looked at so far included the term Bxy. This term leads to a hyperbolic graph: 4xy 8 0 or, solved for y: 8 2 y 4x x What About term Bxy? If there is athe Bxy term: Ax Bxy Cy Dx Ey F 0 2 2 You need to find the discriminant and use that to determine the conic section. B 4 AC 0 The graph is a circle (A = C) or an ellipse (A ≠ C) B 4 AC 0 The graph is a parabola B 4 AC 0 The graph is a hyperbola 2 2 2 Summary ... General Form of a Conic Equation: Ax 2 Bxy Cy 2 Dx Ey F 0 where A, B, C, D, E and F are integers and A, B and C are NOT ALL equal to zero. Identifying a Conic Equation: Conic Parabola Circle Ellipse Hyperbola Equation Stats A = 0 or C = 0, but not both. If A = 0, then the If C = 0, then the parabola is horizontal. parabola is vertical. A=C A & C have the same sign. A & C have different signs. Practice ... Identify each of the following equations as a(n): (a) ellipse (b) circle (d) parabola (e) not a conic (c) hyperbola See if you can rewrite each equation into its Graphing Form! 1) x 2 4y 2 2x 24y 33 0 2) 4x 2 4y 2 9 0 3) x 2 4x y 0 4) x 2 y2 2x 8 0 5) 9x 2 25y 2 54x 50y 119 0 6) x 2 x 0 7) y 2 8y 9x 52 0 8) x 2 2x y 2 4y 7 0 Answers ... (a) ellipse (b) circle (c) hyperbola (d) parabola (e) not a conic 1) x 4y 2x 24y 33 0 2 2 2) 4x 4y 9 0 2 2 (x 1)2 (y 3)2 - -- > (a) 1 4 1 x2 y2 - -- > (c) 9 9 1 4 4 3) x 2 4x y 0 - --> (d) (y 4) (x 2)2 4) x 2 y2 2x 8 0 - -- > (b) (x 1) 2 y2 9 (x 3) 2 (y 1)2 5) 9x 25y 54x 50y 119 0 - - > (a) 1 25 9 6) x 2 x 0 - --> (e) not a conic 2 2 7) y 2 8y 9x 52 0 --- > (d) 9(x 4) (y 4)2 8) x 2x y 4y 7 0 (x 1)2 (y 2)2 - -- > (c) 1 4 4 2 2 Write the general from of the equation fo the translation of -6x2 + 24x + 4y – 8 = 0 for T(-1, -2) 6(x 1) 24(x 1) 4(y 2) 8 0 2 6(x 2x 1) 24(x 1) 4(y 2) 8 0 2 6x 12x 6 24 x 24 4 y 8 8 0 2 6x 12x 4 y 18 0 2 Identify the graph of each equation and then find θ 2x 4 xy y 3 0 2 Use this Formula: 2 B tan 2 A C 37.98 4 tan 2 2 1 tan 1tan2 tan 14 Ellipse 2 75.96 Identify the graph of each equation and then find θ x 3xy 3y 3 2 2 Hyperbola Use this Formula: B tan 2 A C 3 tan 2 1 3 3 tan tan 2 tan 2 1 1 2 56.3 28.15 Identify the graph of each equation and then find θ 3x 4 3xy y 15 2 Use this Formula: 2 B tan 2 A C 4 3 tan 2 3 (1) 1 1 4 3 tan tan 2 tan 4 2 60 Hyperbola 30 Solve this system of equations: 2x y 8 x2 y2 9 Substitution: Straight Line and a circle Not Factorable Step 1 Solve for a variable y 2x 8 Step 2 Plug into other equation x 2 (2x 8) 2 9 x 4 x 32x 64 9 2 2 5x 32x 55 0 2 NO SOLUTION!!! Solve this system of equations: x y 4 2 2 y 1 Substitution: Hyperbola And a straight line Step 1 Solve for a variable x (1) 4 2 2 x 5 2 Step 3 Plug into step 1 to find the other variable y 1 y 1 Step 2 Plug into other equation x 5 Solution(s): ( 5,1) Solve this system of equations: y 2 CRICLE And a 2 2 4 x 9y 36 ELLIPSE x2 y2 4 Step 3 Plug into first equation to find the other variable ELIMINATION: Step 1 Make a new system x 2 (2)2 4 4 x 2 4 y 2 16 x 2 (2)2 4 4 x 2 9y 2 36 Step 2 Combine to eliminate x 4 4 x2 4 4 x 2 0 x2 0 5y 2 20 Solution(s): y 4 2 2 (0, 2) Solve this system of equations: Hyperbola And a 2 2 9x y 16 0 Ellipse x 1 5x 2 y 2 30 Elimination: Step 1 Re-write the system: Step 3 Plug into step 1 to find the other variable 5(1) y 30 2 2 2 5x y 30 5 y 2 30 9x y 16 y 25 2 2 2 2 Step 2 Combine to eliminate 14 x 14 2 x 1 2 y 5 Solution(s): (1, 5) (1, 5) Solve this system of equations: 2y x 3 0 x 2 16 y 2 Substitution: circle And a straight line 5y 12y 7 0 2 12 (12)2 4(5)(7) y 2(5) Step 1 Solve for a variable y x 2y 3 Step 2 Plug into other equation 2y 3 2 16 y 2 4 y 2 12y 9 16 y 2 12 284 0.48 10 y 12 284 2.89 10 Step 3 Plug into step 1 to find the other variable x 2(0.48) 3 x 3.96 x 2(2.89) 3 x 2.78 Solution(s): (3.96, 0.48) and (2.78, 2.89)