CHAT Pre-Clculus Hyperols The third type of conic is clled hyperol. For n ellipse, the sum of the distnces from the foci nd point on the ellipse is fixed numer. For hyperol, the difference of the distnces from the foci nd point on the hyperol is fixed numer. Definition of Hyperol A hyperol is the set of ll points (x, y) in plne, the difference of whose distnces from two distinct fixed points (foci) is positive constnt. (x, y) d d 1 Focus Focus d - d 1 is constnt. 1
CHAT Pre-Clculus Brnch Trnsverse xis c Brnch Focus Center Focus Vertex The midpoint etween the foci is the center. The line segment joining the vertices is the trnsverse xis. The points t which the line through the foci meets the hyperol re the vertices. The grph of the hyperol hs two disconnected rnches. The vertices re units from the center. The foci re c units from the center.
CHAT Pre-Clculus Stndrd Eqution of Hyperol The stndrd form of the eqution of hyperol centered t (h, k) is ( x h) ( y k) 1 Trnsverse xis is horizontl ( y k) ( x h) 1 Trnsverse xis is verticl The vertices re units from the center, nd the foci re c units from the center, with c. If the center is t the origin (0, 0), the eqution tkes one of the following forms. x y 1 Trnsverse xis is horizontl y x 1 Trnsverse xis is verticl 3
CHAT Pre-Clculus ( x h) ( y k) 1 ( y k) ( x h) 1 (h, k+c) (h-c, k) (h, k) (h+c, k) (h, k) (h, k-c) *In generl, if the x-term is listed first, oth rnches of the hyperol will cross the x-xis. If the y-term is listed first, oth rnches of the hyperol will cross the y-xis. Wht out? The vlue of ids us in grphing the hyperol y helping us find the symptotes.
CHAT Pre-Clculus Asymptotes of Hyperol Ech hyperol hs two symptotes tht intersect t the center of the hyperol. The symptotes pss through the vertices of rectngle formed using the vlues of nd nd centered t (h, k). symptote symptote (h, k) There re xes for hyperol. conjugte xis (length = ) (h, k) trnsverse xis (length = ) 5
CHAT Pre-Clculus Looking t the grph, nd using slope = rise/run, we cn see tht the slope of the symptotes is. This will lwys e the cse when the trnsverse xis is horizontl. When the trnsverse xis is verticl the slope =. Asymptotes of Hyperol The eqution for the symptotes of hyperol re y k ( x h) Trnsverse xis is horizontl y k ( x h) Trnsverse xis is verticl If the center of the hyperol is t the origin, then the symptotes re y x Trnsverse xis is horizontl y x Trnsverse xis is verticl 6
Exmple: Find the eqution of the symptotes of the hyperol given y CHAT Pre-Clculus ( x 1) ( y 16 3) 9 1 Solution: The trnsverse xis is horizontl so the eqution is of the form y k ( x h). Sustituting the vlues from our eqution gives us y 3 3 ( x 1) Exmple: Find the eqution of the symptotes of the hyperol given y y 5 x 1 Solution: The center is t the origin nd the trnsverse xis is verticl so the eqution is of the form y x. Sustituting the vlues from our eqution gives 5 us y x. 7
Exmple: Sketch the hyperol given y the eqution CHAT Pre-Clculus ( x ) ( y 5 3) 1 Solution: The center is t (, 3). The hyperol opens right nd left ecuse the x-term is listed first. The trnsverse xis is 10 units, s = 5 The conjugte xis is units, s =. Drw the ox nd sketch the symptotes. Then sketch the hyperol. 8
Exmple: Sketch the hyperol given y the eqution CHAT Pre-Clculus Solution: ( y ) ( x 16 5) 9 1 The center is t (-5, ). The hyperol opens up nd down ecuse the y-term is listed first. The trnsverse xis is 8 units, s = The conjugte xis is 6 units, s = 3. Drw the ox nd sketch the symptotes. Then sketch the hyperol. 9
CHAT Pre-Clculus Exmple: Sketch the grph of the hyperol given y x 9y 9 Solution: Divide through y 9 to get 1 on the right. x x 9 9y y 1 1 9 10
CHAT Pre-Clculus Exmple: Find the stndrd form of the eqution of the hyperol with foci (-1, ) nd (5, ) nd vertices (0, ) nd (, ). Solution: Sketch the given informtion. You will see tht The center must e (, ). The trnsverse xis is horizontl. The stndrd eqution must e of the form ( x h) ( y k) 1 We know tht = nd c = 3, so we cn find. 5 Therefore, our eqution must e c 3 9 ( x ) ( y ) 5 1 11
CHAT Pre-Clculus Exmple: Sketch the grph of the hyperol given y x 9y x 7y 7 0 Solution: Find the eqution y completing the squre. *You must pull out -9 from the y-terms. Be creful. ( x 6x ) 9( y 8y ) 7 *Be creful filling in your lnks on the right side. ( x 6x 9) 9( y ( x 3) 8y 9( y 16) ) 7 36 36 1 *Divide through y -36 nd write in correct form. ( x 3) 36 ( x 3) 9 ( y ) 9( y ) 36 ( y ) ( x 3) 9 1 1 36 36 Sketch the grph. 1
CHAT Pre-Clculus ( y ) ( x 3) 9 1 Exmple: Find the eqution of the hyperol if the vertices re (0,3) nd (0,-3) nd the symptotes re y = 3x nd y = -3x. Solution: Grph the given informtion. By the picture you cn see tht the hyperol opens up nd down, so the symptotes re of the form y x. Since we know = 3 then we must hve = 1. The eqution is y x 1 9 1. 13
CHAT Pre-Clculus Exmple: Find the eqution of the hyperol if the foci re (0,10) nd (0,-10) nd the symptotes re y = ¾ x nd y = -¾ x. Solution: By the picture you cn see tht the hyperol opens up nd down, so the symptotes re of the form y x. Since our eqution is 3 y = " ¾ x, we must hve. Solving gives 3 us. Solve c since we hve c = 10. c 3 10 100 5 16 6 8 3 3 (8) 6 The eqution is ( y 0) 6 ( x 0) y 36 8 x 6 1 1 1
CHAT Pre-Clculus Exmple: Find the eqution of the hyperol with vertices (,1) nd (-,1) nd psses through the point ( 5,0). Solution: We cn tell from the vertices tht =. We cn lso tell tht the center is (0, 1). We know tht the hyperol opens left nd right, so will e of the form ( x h) ( y k) 1 Put in ll of the known informtion nd solve for. ( 5 0) (0 1) 5 1 1 5 1 1 1 1 so 1 ( x 0) ( y 1) The eqution is: 1 15
CHAT Pre-Clculus Grphing Conics Using Grphing Clcultor Since the eqution editor of your grphing clcultor is in the form y =, you will hve to solve ll equtions for y. This my men tking the squre root of oth sides. If tht is the cse, you must grph the positive root in one eqution nd the negtive root in the other. ( y ) ( x 3) Exmple: Grph 1 9 clcultor. ( y ) ( y ) ( y ) y y ( x 3) 9 ( x 3) 9 ( x 3) 9 1 1 ( x 3) 9 ( x 3) 9 1 1 using grphing 1 16
CHAT Pre-Clculus Grph this s the following equtions: ( x 3) ( x 3) y 1 1 nd y 1 9 9 Type the following in your eqution editor: y 1 ((( X 3) /9) 1) y ((( X 3) /9) 1) You my hve to djust your viewing window. 17
CHAT Pre-Clculus Eccentricity Definition of Eccentricity The eccentricity e of hyperol is given y the rtio e *Note: Becuse c is lwys greter thn, the frction c will lwys e greter thn 1. If the eccentricity is close to 1, the rnches of the hyperol re more pointed. If the eccentricity is gret, the rnches re fltter. c Exmple: Find the eccentricity of the hyperol given y ( y ) ( x 3) 9 1 13 Solution: Solving for c gives us e 1. 8 c 18
Generl Equtions of Conics The grph of Ax + Cy + Dx + Ey + F = 0 is one of the following: 1. A circle if A = C CHAT Pre-Clculus. A prol if AC = 0 (A=0 or C=0, ut not oth) 3. An ellipse if AC > 0 (A nd C hve like signs). A hyperol if AC < 0 (A nd C hve unlike signs) Exmple: Clssify ech of the following. ) x + 5y 9x + 8y = 0 ellipse (AC = 0) ) x 5x + 7y 8 = 0 prol (AC = 0) c) 7x + 7y 9x + 8y - 16 = 0 circle (A = C) d) x - 5y x + 8y + 1 = 0 hyperol (AC = -0) 19