1 LC The Hyperbola MCR3U - Santowski

2 (A) Hyperbola as Loci A hyperbola is defined as the set of points such that the difference of the distances from any point on the hyperbola to two stationary points (called the foci) is a constant A hyperbola is defined as the set of points such that the difference of the distances from any point on the hyperbola to two stationary points (called the foci) is a constant We will explore the hyperbola from a locus definition in two ways We will explore the hyperbola from a locus definition in two ways ex 3. Using grid paper with 2 sets of concentric circles, we can define the two circle centers as fixed points and then label all other points (P), that meet the requirement that the difference of the distances from the point (P) on the hyperbola to the two fixed centers (which we will call foci) will be a constant i.e. | PF 1 - PF 2 | = constant. We will work with the example that PF 1 - PF 2 = 4 units. ex 3. Using grid paper with 2 sets of concentric circles, we can define the two circle centers as fixed points and then label all other points (P), that meet the requirement that the difference of the distances from the point (P) on the hyperbola to the two fixed centers (which we will call foci) will be a constant i.e. | PF 1 - PF 2 | = constant. We will work with the example that PF 1 - PF 2 = 4 units. ex 4. Using the GSP program, we will geometrically construct a set of points that satisfy the condition that | PF 1 - PF 2 | = constant by following the directions on the handout ex 4. Using the GSP program, we will geometrically construct a set of points that satisfy the condition that | PF 1 - PF 2 | = constant by following the directions on the handout

3 (B) Hyperbolas as Loci

4 (C) Hyperbolas as Loci - Algebra We will now tie in our knowledge of algebra to come up with an algebraic description of the hyperbola by making use of the relationship that | PF 1 - PF 2 | = constant We will now tie in our knowledge of algebra to come up with an algebraic description of the hyperbola by making use of the relationship that | PF 1 - PF 2 | = constant ex 5. Find the equation of the hyperbola whose foci are at (+3,0) and the constant (which is called the difference of the focal radi) is 4. Then sketch the hyperbola by finding the x and y intercepts. ex 5. Find the equation of the hyperbola whose foci are at (+3,0) and the constant (which is called the difference of the focal radi) is 4. Then sketch the hyperbola by finding the x and y intercepts.

5 (C) Hyperbolas as Loci - Algebra Since we are dealing with distances, we set up our equation using the general point P(x,y), F 1 at (-3,0) and F 2 at (3,0) and the algebra follows on the next slide |PF 1 - PF 2 | = 4 Since we are dealing with distances, we set up our equation using the general point P(x,y), F 1 at (-3,0) and F 2 at (3,0) and the algebra follows on the next slide |PF 1 - PF 2 | = 4

6 (C) Algebraic Work

7 (D) Graph of the Hyperbola

8 (E) Analysis of the Hyperbola The equation is (x/2) 2 - (y/  5) 2 = 1 OR 5x 2 - 4y 2 = 20 The equation is (x/2) 2 - (y/  5) 2 = 1 OR 5x 2 - 4y 2 = 20 The x-intercepts occur at (+2,0) and there are NO y-intercepts The x-intercepts occur at (+2,0) and there are NO y-intercepts The domain is {x E R | -2 > x > 2} and the range is {y E R } The domain is {x E R | -2 > x > 2} and the range is {y E R } NOTE that this is NOT a function, but rather a relation NOTE that this is NOT a function, but rather a relation NOTE the relationship between the equation and the intercepts, domain and range  so to generalize, if the hyperbola has the standard form equation (x/a) 2 - (y/b) 2 = 1, then the x-intercepts occur at (+a,0), and the domain is {x E R | -a > x > a} and the range is any real number NOTE the relationship between the equation and the intercepts, domain and range  so to generalize, if the hyperbola has the standard form equation (x/a) 2 - (y/b) 2 = 1, then the x-intercepts occur at (+a,0), and the domain is {x E R | -a > x > a} and the range is any real number We can rewrite the equation in the form of (bx) 2 - (ay) 2 = (ab) 2 We can rewrite the equation in the form of (bx) 2 - (ay) 2 = (ab) 2 Note that if the equation is (x/a) 2 - (y/b) 2 = 1, then the hyperbola opens along the x-axis Note that if the equation is (x/a) 2 - (y/b) 2 = 1, then the hyperbola opens along the x-axis BUT if the equation is (y/a) 2 - (x/b) 2 = 1 (OR (x/b) 2 - (y/a) 2 = -1) then the hyperbola opens along the y-axis BUT if the equation is (y/a) 2 - (x/b) 2 = 1 (OR (x/b) 2 - (y/a) 2 = -1) then the hyperbola opens along the y-axis

9 (E) Analysis of the Hyperbola This hyperbola has the form (x/b) 2 – (y/a) 2 = -1 (or (y/a) 2 – (x/b) 2 = 1 This hyperbola has the form (x/b) 2 – (y/a) 2 = -1 (or (y/a) 2 – (x/b) 2 = 1

10 (E) Analysis of the Hyperbola To get the correct shape of the hyperbola, we need to find and graph the asymptotes of the hyperbola  the fact that a hyperbolic curve approaches asymptotes is what makes it different from a parabola To get the correct shape of the hyperbola, we need to find and graph the asymptotes of the hyperbola  the fact that a hyperbolic curve approaches asymptotes is what makes it different from a parabola For hyperbolas that open along the x-axis, the equation of the asymptotic lines are y = +(b/a)x For hyperbolas that open along the x-axis, the equation of the asymptotic lines are y = +(b/a)x To easily graph the asymptotes on the graph, we simply form a rectangular box using the values of a and b (corners of the box are (a,b), (-a,b), (-a,-b), (a,-b)) To easily graph the asymptotes on the graph, we simply form a rectangular box using the values of a and b (corners of the box are (a,b), (-a,b), (-a,-b), (a,-b))

11 (E) Analysis of the Hyperbola The axis upon which the hyperbola opens is called the major axis and lies between the 2 x-intercepts. Its length is 2a (vice versa if direction of opening is along y-axis) The axis upon which the hyperbola opens is called the major axis and lies between the 2 x-intercepts. Its length is 2a (vice versa if direction of opening is along y-axis) The two end points of the major axis (in this case the x- intercepts) are called vertices (at (+a,0)) The two end points of the major axis (in this case the x- intercepts) are called vertices (at (+a,0)) The two foci lie on the major axis The two foci lie on the major axis The domain is {x E R | -a > x > a} and the range is any real number The domain is {x E R | -a > x > a} and the range is any real number The asymptotes are at y = + (b/a)x (BUT if the hyperbola opens up/down, then the equation of the asymptotes is y = + (a/b)x) The asymptotes are at y = + (b/a)x (BUT if the hyperbola opens up/down, then the equation of the asymptotes is y = + (a/b)x)

12 (F) In-class Examples Determine the equation of the hyperbola and then sketch it, labelling the key features, if the foci are at (0,+5) and the difference of the focal radii is 6 units (i.e.  the fancy name for the constant distance sum PF 1 - PF 2 ) Determine the equation of the hyperbola and then sketch it, labelling the key features, if the foci are at (0,+5) and the difference of the focal radii is 6 units (i.e.  the fancy name for the constant distance sum PF 1 - PF 2 ) The equation you generate should be either The equation you generate should be either y 2 /9 - x 2 /16 = 1 y 2 /9 - x 2 /16 = 1 or x 2 /16 – y 2 /9 = -1 or x 2 /16 – y 2 /9 = -1

13 (G) Internet Links a/EquationHyperbola.html - an interactive applet fom AnalyzeMath a/EquationHyperbola.html - an interactive applet fom AnalyzeMath a/EquationHyperbola.html a/EquationHyperbola.html - Examples and explanations from OJK's Precalculus Study Page - Examples and explanations from OJK's Precalculus Study Page /Hyperbolas.asp - Ellipses from Paul Dawkins at Lamar University /Hyperbolas.asp - Ellipses from Paul Dawkins at Lamar University /Hyperbolas.asp /Hyperbolas.asp - Graphs of ellipses from WebMath.com - Graphs of ellipses from WebMath.com

14 (G) Homework AW, p481, Q7ab, 8ab AW, p481, Q7ab, 8ab Nelson text, p616, Q7,15,16,6 Nelson text, p616, Q7,15,16,6