SalasSV_09_01

# SalasSV_09_01 - CHAPTER 9 Translations THE CONIC SECTIONS...

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CHAPTER 9 THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS ± 9.1 TRANSLATIONS; THE PARABOLA Translations In Figure 9.1.1 we have drawn a rectangular coordinate system and marked a point O ± ( x 0 , y 0 ). Think of the Oxy system as a rigid frame and in your mind slide it along the plane, without letting it turn, so that the origin O falls on the point O ± (Figure 9.1.2). Such a move, called a translation , produces a new coordinate system O ± XY . y x O O ( x 0 , y 0 ) Figure 9.1.1 Y X O y x O Figure 9.1.2 A point P now has two pairs of coordinates: a pair ( x , y ) with respect to the Oxy system and a pair ( X , Y ) with respect to the O ± XY system (Figure 9.1.3). To see the relation between these coordinates, note that, starting at O , we can reach P by ±rst going to O ± and then going on to P ; thus x = x 0 + X , y = y 0 + Y . Translations are often used to simplify geometric arguments. To illustrate this, we will derive a formula for the distance between a point and a line. 514

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9.1 TRANSLATIONS; THE PARABOLA ± 515 y x O y 0 Y P x 0 X y x O Figure 9.1.3 Let l be a line and let P be a point that is not on l . It is easy to see that the point Q on l that is closest to P is the foot of the perpendicular from P to l . See Figure 9.1.4. The distance between P and l , denoted d ( P , l ), is de±ned to be d ( P , Q ). y x P Q R l Figure 9.1.4 9.1.1 DISTANCE BETWEEN A POINT AND A LINE The distance between the line l : Ax + By + C = 0 and the point P 1 ( x 1 , y 1 )is given by the formula d ( P 1 , l ) = | Ax 1 + By 1 + C | A 2 + B 2 . PROOF First we ±nd the distance between the origin O and a line l : Ax + By + C = 0. We assume B ±= O and leave the case B = O to you. Since l has slope A / B , the line through the origin perpendicular to l has slope B / A and equation y = B A x , which we write as Bx Ay = 0. Solving the equations Ax + By + C = 0 Bx = 0
516 ± CHAPTER 9 THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS simultaneously, we fnd that x = AC A 2 + B 2 , y = BC A 2 + B 2 . Thus, the Foot oF the perpendicular From l to the origin is the point Q with coordinates ± A 2 + B 2 , BC A 2 + B 2 ² (±igure 9.1.5) ThereFore d ( O , l ) = d ( O , Q ) = ³ ´ A 2 + B 2 µ 2 + ´ BC A 2 + B 2 µ 2 = | C | A 2 + B 2 . y x l Q Figure 9.1.5 Now let P 1 ( x 1 , y 1 ) be an arbitrary point in the plane. We translate the Oxy coordinate system to obtain a new coordinate system O ± XY with O ± Falling on P 1 (see ±igure 9.1.6). The new coordinates are related to the old coordinates as Follows: x = x 1 + X , y = y 1 + Y . In the xy -system, l has equation Ax + By + C = 0. In the XY -system, l has equation A ( x 1 + X ) + B ( y 1 + Y ) + C = 0. We can write this last equation as AX + BY + K = 0 with K = Ax 1 + By 1 + C . y x P 1 ( x 1 , y 1 ) Y X l Figure 9.1.6 The distance we want is the distance between the line AX + BY + K = 0 and the new origin O ± . As noted above, this distance is | K | A 2 + B 2 . Since K = Ax 1 + By 1 + C ,wehave d ( P 1 , l ) = | Ax 1 + By 1 + C | A 2 + B 2 . ± Example 1 ±ind the distance between the line l :3 x + 4 y 5 = 0 and (a) the origin, (b) the point P ( 6, 2).

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SalasSV_09_01 - CHAPTER 9 Translations THE CONIC SECTIONS...

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