EllipsesEllipses andand HyperbolasHyperbolas In this chapter we'll see three more examples of conics. In this chapter we’ll revisit some examples of conies. EllipsesEllipses 1 If youIf beginyou begin withwith the unitthe unit circle,circle.C ,C’. andand youyou scalescalex-coordinatesr-coordmates byby somesome nonzerononzero numbernumbera, anda. youand you scalescaley-coordinatesj-coordinates byby somesome nonzerononzero numbernumber bb,, - the resultingthe resulting shapeshape in thein planethe plane is called- is called an anellipseellipse.. on on IoI I-o I I-o ‘I I II Let’s begin again with the unit circle ‘I II C’, the circle of radius 1 centered at Let's begin again with the unit circle C1, the circle of radius 1 centered at (0, 0). We learned earlier that C’ is the set of solutions of the equation (0; 0). We learned earlier that C1 is the set of solutions of the equation x-+y9 2 =1 x2 + y2 = 1 The matrixa 0D scales the x-coordinates in the plane by a, and The matrix scales= ( the) x-coordinates in the plane by a, and it scales it scales the0 y-coordinatesb by b. What’s drawn below is D(C’), which is the shape resulting from scaling C’ horizontally by a and verticallyaby 0b.It’s an the y-coordinates by b. What's drawn below on the right is (C1), example of an ellipse. 0 b which is the shape resulting from scaling C1 horizontallyH’ by a and vertically by b. It's an example of an ellipse. op 0 op Using POTS, the equation for this distorted circle, the ellipse D(C’), is obtained by precomposing the equation for C’, the equation x2 + y2 = 1, by the matrix op _ z:3 -co op _ z:3 D1=(j -co II + + ?) II + + 1874 4 0 0 II II + 204 + - - on on - I-o I I-o ‘I I II ‘I II on I-o I ‘I a 0 Using POTS,II the equation for this distorted circle, the ellipse (C1), Ellipses and Hyperbolas 0 b 1 2 2 is obtainedIn this bychapter precomposingwe’ll revisit the equationsome examples for Cof,conies. the equation x + y = 1, by the matrix Ellipses −1 1 a 0 a 0 If you begin with the unit circle. =C’. and1 you scale r-coordmates by some 0 b 0 b nonzero number a. and you scale j-coordinates by some nonzero number b, the resulting shape in the plane is called an ellipse. op op op _ z:3 -co op _ z:3 -co op II + IoI+ II + + Let’s begin again with the unit circle4 C’, the circle of radius 1 centered at 4 0 0 (0, 0). We learned earlier that C’ is the setII of solutions of the equation II 9 2 + x-+y =1 + op _ z:3 The matrix D scales the x-coordinates-co in the plane by a, and II + + = ( ) 4 it scales the y-coordinates by b. What’s drawn below is D(C’), which is the 0 shape resulting from scaling C’ horizontallyII by a and vertically by b. It’s an example of an ellipse. + H’ 1 a 0 x y Because 1 replaces x with a and y with b , the equation for the ellipse 0 b a 0 (C1) is 0 b 0 x2 y2 + = 1 Using POTS, the a b which is equivalent to theequation equationfor this distorted circle, the ellipse D(C’), is obtained by precomposing the equation for C’, the equation x2 + y2 = 1, by the matrix x2 y2 2 + 2 = 1 a D1=(jb ?) 187 The equation for an ellipse described above is important. It's repeated at the top of the next page. 205 Ellipses and Hyperbolas Suppose a; b 2 − f0g. The equation for the ellipse In this chapter we’ll revisit some examples ofR conies. obtained by scaling the unit circle Ellipses by a in the x-coordinate and by b in the y-coordinate is If you begin with the unit circle. C’. and you scalex2r-coordmatesy2 by some + = 1 nonzero number a. and you scale j-coordinates by asome2 bnonzero2 number b, the resulting shape in the plane is called an ellipse. IoI Let’s begin again with the unit circle C’, the circle of radius 1 centered at (0, 0). We learned earlier that C’ is the set of solutions of the equation x-+y9 2 =1 Example. The matrix D scales the x-coordinates in the plane by a, and x2 y2 = • The equation for the ellipse shown below is + = 1. In this ( ) 25 4 it scales the y-coordinatesexample,by web. areWhat’s usingdrawn the formulabelow is fromD(C’), thewhich top ofis thethe page with a = 5 and shape resulting fromb = 2.scaling C’ horizontally by a and vertically by b. It’s an example of an ellipse. H’ 0 tn Using POTS, the equation for this distorted circle, the ellipse D(C’), is obtained by precomposing the equation for C’, the equation x2 + y2 = 1, by the matrix ************* D1=(j ?) 187 206 Ellipses and Hyperbolas In this chapter we’ll revisit some examples of conies. Ellipses If you begin with the unit circle. C’, and ou scale i-coordinates by some nonzero number a, and you scale y-coordinates by some nonzero number b, the resulting shape in the plane is called an ellipse. C, b r Let’s begin again with the unit circle the circle of radius 1 centered at (0. 0). We learned earlier that C’ is the set of solutions of theEllipsesequation and Hyperbolas 9 9 In this chapter we’ll revisit some examples of conies. Circles are ellipses The matrix D scal:s th:x-coordinates in the plane by a, and A circle is a perfectly round ellipse. Ellipses = ( ) If weit scales scalethe they-coordinates unit circle byby theb. What’s same numberdrawnIfbelowryou > begin0is inD(C’), bothwith which coordinates,the unitis thecircle. C’, and ou scale i-coordinates by some thenshape we'll stretchresulting thefrom unitscaling circleC’ evenlyhorizontally in all directions.nonzeroby a andnumberverticallya, byandb. youIt’s scalean y-coordinates by some nonzero number b, example of an ellipse. the resulting shape in the plane is called an ellipse. C, C’ b Ellipses and Hyperbolas r In this chapter we’ll revisit some examples of conies. Ellipses If you begin with the unit circle. C’. and you scale r-coordmates by some Using POTS. the equation for this distortedLet’scircle,begintheagainellipsewithD(C’).the unitis circle the circle of radius 1 centered at nonzero number a. Theand resultyou scale willj-coordinates be another circle,by some onenonzero whosenumber equationb, is obtained by precomposing the equation for C1,(0. 0).the Weequationlearnedi2earlier+ y2 =that1, byC’ is the set of solutions of the equation the resulting shape in the plane is called an ellipse. the matrix x2 y2 9 9 + = 1 rD’=(E52 r2 We can multiply both sides of this equation) byTher2 tomatrix obtainD the equivalentscal:s th:x-coordinates in the plane by a, and 187 = ( ) equation it scales the y-coordinates by b. What’s drawn below is D(C’), which is the 2 2 2 x + y = r shape resulting from scaling C’ horizontally by a and vertically by b. It’s an which we had seen earlier as the equation for Crexample, the circleof an ofellipse. radius r centered at (0; 0). IoI C’ Let’s begin again with the unit circle C’, the circle of radius 1 centered at (0, 0). We learned earlier that*************C’ is the set of solutions of the equation x-+y9 2 =1 The matrix D Hyperbolasscales the fromx-coordinates scalingin the plane by a, and = We've( ) seen one example of a hyperbola, namely the set of solutions of the it scales the y-coordinates by b. What’s drawn below is D(C’), which is the 1 Using POTS. the equation for this distorted circle, the ellipse D(C’). is shape resulting fromequationscalingxyC’=horizontally 1. We'll callby thisa and hyperbolaverticallyHby ,b. theIt’sunitan hyperbola. obtained by precomposing the equation for C1, the equation i2 + y2 = 1, by example of an ellipse. the matrix H’ D’=(E5 ) 0 187 Using POTS, the equation for this distorted circle, the207ellipse D(C’), is obtained by precomposing the equation for C’, the equation x2 + y2 = 1, by the matrix D1=(j ?) 187 Ellipses and Hyperbolas In this chapter we’ll revisit some examples of conies. Ellipses If you begin with the unit circle. C’. and you scale r-coordmates by some nonzero number a. and you scale j-coordinates by some nonzero number b, the resulting shape in the plane is called an ellipse. IoI Let’s begin again with the unit circle C’, the circle of radius 1 centered at 1 (0, learned Suppose c > 0. We can horizontally stretch the unit hyperbola H using 0). We earlier that C’ is the set of solutions of the equation 9 c 0 x-+y 2 =1the diagonal matrix . This is the matrix that scales the x-coordinate 0 1 by c and does not alter the y-coordinate. The resulting shape in the plane, The matrix D scales the x-coordinates in the plane by a, and = c 0 ( ) (H1), is also called a hyperbola. it scales the y-coordinates by b. What’s drawn0 1below is D(C’), which is the shape resulting from scaling C’ horizontally by a and vertically by b. It’s an example of an ellipse. - H’ - - (c,i) .xI—cx on D(H) on 0 on Using POTS, the equation for this distorted circle, the ellipse D(C’), ) is ,)ço I-o I ‘4O obtained by precomposing = ‘I the equation for C’, the equation x2 1, by I-o I-o II + y2 I I 1 o fj ‘I c\ ‘I II II the matrix =) D1=(j ?) 187 Jo c 0 The equation for the hyperbola (H1) is obtained by precomposing 0 1 c 0−1 1 0 the equation for H1, the equation xy = 1, with = c .