Analytic Geometry Lecture Note

Analytic Geometry 01/08/24 Distance between two points and division of line segment Rectangular (x,y) distance formula d = sqrt((x2-x1)^2 + (y2-y1)^2) i) Y (x1,y1) d (x2,y2) X Ex: sqrt((2-(-4))^2 + (0-2)^2) = sqrt(6^2 + (-2)^2) = sqrt(36+4) = sqrt(40) = 2 sqrt(10) A (-4,2) and B (2,0) x1 = -4, y1 = 2 x2 = 2, y2 = 0 A (1,2) B (7,6) midpoint xm = (x1+x2)/2 = (1+7)/2 = 8/2 = 4 ym = (y1+y2)/2 = (2+6)/2 = 8/2 = 4 (4,4) O (3,4) A (xm,ym) B Y (x,y) r y theta x X r cos theta r sin theta P (-7,5) x = r cos theta y = r sin theta x = -7, y = 5 r = sqrt((-7)^2 + 5^2) = sqrt(49+25) = sqrt(74) tan theta = y/x = 5/-7 theta = arctan(-5/7) (-r, -theta) (- sqrt(74), arctan(5/7)) (-x, -y) (-(-7), -5) = (7,-5) (-r cos theta, -r sin theta) (- sqrt(74) cos(arctan(-5/7)), - sqrt(74) sin(arctan(-5/7))) (- sqrt(74) * (-7/sqrt(74)), - sqrt(74) * (5/sqrt(74))) = (7,-5) (-x, y) (-(-7), 5) = (7,5) (-r cos theta, r sin theta) (- sqrt(74) * (-7/sqrt(74)), sqrt(74) * (5/sqrt(74))) = (7,5) (x, -y) (-7, -5) (r cos theta, -r sin theta) (sqrt(74) * (-7/sqrt(74)), - sqrt(74) * (5/sqrt(74))) = (-7,-5) Equation of Line General form Ax + By + C = 0 Slope-intercept form y = mx + b Point-slope form (y-y1) = m(x-x1) Situation (1-6) A (5,1) B (2,6) 5) Soln 1: (Conventional) m = (y2-y1)/(x2-x1) = (6-1)/(2-5) = 5/-3 = -5/3 y - y1 = m(x - x1) y - 1 = -5/3 (x - 5) y - 1 = -5/3 x + 25/3 y = -5/3 x + 25/3 + 1 y = -5/3 x + 28/3 5x + 3y - 28 = 0 Soln 2: (Concise formula when two points are given) (y-y1)/(x-x1) = (y2-y1)/(x2-x1) (y-1)/(x-5) = (6-1)/(2-5) (y-1)/(x-5) = 5/-3 -3(y-1) = 5(x-5) -3y + 3 = 5x - 25 5x + 3y - 28 = 0 Soln 3: (y-intercept form when one point and gradient are given) y = m(x-x1) + y1 y = -5/3 (x-5) + 1 y = -5/3 x + 25/3 + 1 y = -5/3 x + 28/3 3y = -5x + 28 5x + 3y - 28 = 0 Soln 4: (Concise formula when y-intercept and gradient are given) y = mx + c c = y1 - mx1 = 1 - (-5/3)(5) = 1 + 25/3 = 28/3 y = -5/3 x + 28/3 3y = -5x + 28 5x + 3y - 28 = 0 Soln 5: (Concise formula when x-intercept and gradient are given) y = m(x-d) d = x1 - y1/m = 5 - 1/(-5/3) = 5 + 3/5 = 28/5 y = -5/3 (x - 28/5) y = -5/3 x + 28/3 3y = -5x + 28 5x + 3y - 28 = 0 Soln 6: (Concise formula when intercepts are given on to coordinate axes) x/a + y/b = 1 a = x1 - y1 * (x2-x1)/(y2-y1) = 5 - 1 * (2-5)/(6-1) = 5 - 1 * (-3)/5 = 5 + 3/5 = 28/5 b = y1 - x1 * (y2-y1)/(x2-x1) = 1 - 5 * (6-1)/(2-5) = 1 - 5 * 5/-3 = 1 + 25/3 = 28/3 x/(28/5) + y/(28/3) = 1 5x/28 + 3y/28 = 1 5x + 3y = 28 5x + 3y - 28 = 0 y-intercept (0,4) A (5,6) B (2,0) x y 5 6 2 0 (x1,y1) (x2,y2) m = (6-0)/(5-2) = 6/3 = 2 y = mx + c y = 2x + 4 x-intercept: set y=0 0 = 2x + 4 2x = -4 x = -2 (-2,0) x-intercept Distance from a point to a line and between two lines Situation (7-8) (x0,y0) = (-7,-6) add on What is the slope of the line Offset Ax + By + C = 0 (not used) By = -Ax + C y = -A/B x + C/B (slope intercept form) m = -A/B Ex: 3x - 4y + 5 = 0 m = -3/-4 = 3/4 d = |Ax0 + By0 + C| / sqrt(A^2 + B^2) d = |3(-7) - 4(-6) + 5| / sqrt(3^2 + (-4)^2) d = |-21 + 24 + 5| / sqrt(9 + 16) d = |8| / sqrt(25) = 8/5 Parallel to 2x - 3y - 5 = 0 l1 : m1 l2 : m2 l1 || l2 => m1 = m2 m1 = -2/-3 = 2/3 m2 = 2/3 y - y1 = m(x - x1) Soln 1: (Conventional approach) y - (-7) = 2/3 (x - (-6)) y + 7 = 2/3 (x + 6) 3(y + 7) = 2(x + 6) 3y + 21 = 2x + 12 2x - 3y - 9 = 0 Soln 2: (Concise long parallel) Ax + By = C' 2x - 3y = C' set (x,y) = (-6,-7) 2(-6) - 3(-7) = C' -12 + 21 = C' C' = 9 2x - 3y = 9 2x - 3y - 9 = 0 Perpendicular to 2x + 5y + 7 = 0 l1 : m1 l2 : m2 l1 | l2 => m1 * m2 = -1 => m2 = -1/m1 m1 = -2/5 m2 = -1 / (-2/5) = 5/2 (reciprocal) m2 = 5/2 y - y1 = m(x - x1) y - (-7) = 5/2 (x - (-6)) y + 7 = 5/2 (x + 6) 2(y + 7) = 5(x + 6) 2y + 14 = 5x + 30 5x - 2y + 16 = 0 Distance from a point to a line and between two lines Situation (9-10) 9) (x1, y1) d Ax + By + C = 0 direct shortest distance distance perpendicular to the line (x0, y0) = (6,9) 3x - 5y + 12 = 0 d = |Ax0 + By0 + C| / sqrt(A^2 + B^2) d = |3(6) - 5(9) + 12| / sqrt(3^2 + (-5)^2) d = |18 - 45 + 12| / sqrt(9 + 25) d = |-15| / sqrt(34) = 15 / sqrt(34) = 15 sqrt(34) / 34 ≈ 2.57 x + 3y - 9 = 0 d Ax + By + C1 = 0 Ax + By + C2 = 0 d = |C2 - C1| / sqrt(A^2 + B^2) d = |-9 - (-10)| / sqrt(1^2 + 3^2) d = |1| / sqrt(10) = 1 / sqrt(10) = sqrt(10) / 10 ≈ 0.32 x + 3y - 10 = 0 two parallel lines; distance between parallel para matinding linya x + 3y = 9 x + 3y = 10 para same A at B d = |10 - 9| / sqrt(1^2 + 3^2) = 1 / sqrt(10) Locus of Points Situation (11-12) equal distance = equal distance d1 = d2 11) Y (x,y) (4,5) d2 d1 (2,1) X sqrt((x-4)^2 + (y-5)^2) = sqrt((x-2)^2 + (y-1)^2) (x-4)^2 + (y-5)^2 = (x-2)^2 + (y-1)^2 x^2 - 8x + 16 + y^2 - 10y + 25 = x^2 - 4x + 4 + y^2 - 2y + 1 -8x - 10y + 41 = -4x - 2y + 5 -4x - 8y + 36 = 0 4x + 8y = 36 x + 2y = 9 y = 4 Y (x,y) d2 y = 4 d1 (0,0) X d1 = sqrt((x-0)^2 + (y-0)^2) = sqrt(x^2 + y^2) d2 = |y - 4| sqrt(x^2 + y^2) = |y - 4| x^2 + y^2 = (y - 4)^2 x^2 + y^2 = y^2 - 8y + 16 x^2 = -8y + 16 x^2 = -8(y - 2) Conic Sections Parabola Circle Ellipse Hyperbola U O O >< General Form Ax 2 +Bxy+Cy 2 +Dx+Ey+F=0 Discriminant B 2 −4AC B 2 −4AC<0 B 2 −4AC=0 B 2 −4AC>0 Shape Ellipse Parabola Hyperbola Condition A = C A = 0 or C = 0 Ax^2 and Cy^2 have same sign Ax^2 and Cy^2 have opposite sign Shape Circle Parabola Ellipse Hyperbola Circle Standard form (x−h) 2 +(y−k) 2 =r 2 (center (h,k)) (5,-3) and (-1,5) by calc. backsolve m) lahat ng choices gamit isa sa mga coordinates C) x 2 +y 2 −4x−2y−20=0 general to standard form x 2 −4x+y 2 −2y=20 (x 2 −4x+4)+(y 2 −2y+1)=20+4+1 (x−2) 2 +(y−1) 2 =25 -> standard form center (h,k) = (2,1) radius r = sqrt(25) = 5 Challenge Problem: (x+2) 2 +(y+5) 2 =36 center = (-2,-5) r = sqrt(36) = 6 Y r = 10ft (0,6) C (h,k) = (0,6) (x-h)^2 + (y-k)^2 = r^2 (x-0)^2 + (y-6)^2 = 10^2 x^2 + (y-6)^2 = 100 (0,0) X (7,?) x = 7 (7)^2 + (y-6)^2 = 100 49 + (y-6)^2 = 100 (y-6)^2 = 51 y - 6 = +/- sqrt(51) y = 6 +/- sqrt(51) y ≈ 6 +/- 7.14 y1 ≈ 13.14 y2 ≈ -1.14 (7, -1.14) Parabola (along center) Standard form (y−k) 2 =4a(x−h) -> opens right or left (x−h) 2 =4a(y−k) -> opens up or down focus directrix distance of vertex to directrix is equal to focal length Situation (16-18) (y−0) 2 =8(x+6) -> standard form 16) Vertex (h,k) = (-6,0) (h,k) = (-6,0) Focus (x,y) = (-6+2, 0) = (-4,0) (x,y) = (-4,0) 17) axis of symmetry: y-axis or y = 0 equation of directrix: x = -8 or x + 8 = 0 x = -6 - 2 = -8 18) Latus Rectum = |4a| = |8| = 8 Situation (19-20) 19) (x−h) 2 =4a(y−k) (x−0) 2 =4a(y−10) (2100) 2 =4a(500−10) 4410000=4a(490) a=4410000/(4∗490)=4410000/1960=2250 x 2 =4(2250)(y−10) x 2 =9000(y−10) y=x 2 /9000+10 set x = 720 y=(720) 2 /9000+10=518400/9000+10=57.6+10=67.6 Soln 2: (may 3 points na alam sa parabola) UNDER 3-3 (y = Ax^2 + C) x y A 0 10 2100 500 -2100 500 10=A(0) 2 +C=>C=10 500=A(2100) 2 +10 490=A(4410000) A=490/4410000=1/9000 y=(1/9000)x 2 +10=x 2 /9000+10 y=67.6 Soln 2: y=67.6 Ellipse Important Reminders • foci always lie on the major axis • vertices are the endpoints of the major axis while co-vertices are the endpoints of the minor axis • a is always along major axis • b is always along the minor axis • for ellipse, c is always smaller than both a and b c = focal length a = length of the semi-major axis 2a = length of the major axis b = semi-minor axis d1 + d2 = 2a CNC para malaman na ellipse Hyperbola Important Reminders • the transverse axis always intersects the hyperbola • the foci always lie on the transverse axis • a is always along the transverse axis while b is along the conjugate axis • for hyperbola, c is larger than both a and b and a does not necessarily need to be greater than b d1 - d2 = +/- 2a CNC hint for hyperbola ELLIPSE vs. HYPERBOLA ELLIPSE PROPERTIES HYPERBOLA c = sqrt(a^2 - b^2) (first) Eccentricity c = sqrt(a^2 + b^2) (first) e = c/a (second) e = c/a (second) c^2 = a^2 - b^2 Focal Length Eq. c^2 = a^2 + b^2 LR = 2b^2 / a Latus Rectum LR = 2b^2 / a d = a/e Distance fr. center to directrix d = a/e Situation (21-23) (y−3) 2 /25+(x+4) 2 /9=1 21) a 2 =25=>a=5 b 2 =9=>b=3 c 2 =a 2 −b 2 =25−9=16=>c=4 22) e=c/a=4/5=0.8 23) d=a/e=5/(4/5)=25/4=6.25 Center (h,k) = (-4,3) Foci F1 (h, k+c) = (-4, 3+4) = (-4,7) F2 (h, k-c) = (-4, 3-4) = (-4,-1) Vertices V1 (h, k+a) = (-4, 3+5) = (-4,8) V2 (h, k-a) = (-4, 3-5) = (-4,-2) Co-vertices Cv1 (h+b, k) = (-4+3, 3) = (-1,3) Cv2 (h-b, k) = (-4-3, 3) = (-7,3) major axis along y 2a = 10 minor axis along x 2b = 6 SITUATION (24-26) 24) e = 0.248 a = 39.482 AU perihelion = a(1-e) = 39.482(1-0.248) = 39.482(0.752) = 29.696 AU aphelion = a(1+e) = 39.482(1+0.248) = 39.482(1.248) = 49.273 AU 2a = ? => diameter b = ? e = c/a => c = ea = 0.248 * 39.482 = 9.7915 ≈ 9.792 AU a 2 =b 2 +c 2 (39.482) 2 =b 2 +(9.792) 2 1558.839524=b 2 +95.883264 b 2 =1558.839524−95.883264=1462.95626 b= 1462.95626 ​ ≈38.25AU c = 9.792 AU x = 2a - 29.696 x = 2(39.482) - 29.696 = 78.964 - 29.696 = 49.268 AU ≈ 49.3 AU HYPERBOLA Y (h,k) = (-2,0) center focus (0,-5) c = sqrt((0-(-2))^2 + (-5-0)^2) = sqrt(2^2 + (-5)^2) = sqrt(4+25) = sqrt(29) ≈ 5.4 transverse axis along y a = 4 c 2 =a 2 +b 2 ( 29 ​ ) 2 =4 2 +b 2 29=16+b 2 b 2 =13=>b= 13 ​ ≈3.6 standard equation (y−k) 2 /a 2 −(x−h) 2 /b 2 =1 (y−0) 2 /4 2 −(x−(−2)) 2 /( 13 ​ ) 2 =1 y 2 /16−(x+2) 2 /13=1 slope of the asymptotes +/- a/b = +/- 4 / sqrt(13) = +/- 4 sqrt(13) / 13 SITUATION (27-28) (navigation) -> station = focus by using hyperbola 27) ∣dA−dB∣=100 2a=100=>a=50 center (0,0) foci (+/- c, 0) c=200 c 2 =a 2 +b 2 200 2 =50 2 +b 2 40000=2500+b 2 b 2 =37500=>b= 37500 ​ =193.65 equation (x−h) 2 /a 2 −(y−k) 2 /b 2 =1 x 2 /50 2 −y 2 /(193.65) 2 =1 x 2 /2500−y 2 /37500=1 set y = 80 min x 2 /2500−(80) 2 /37500=1 x 2 /2500−6400/37500=1 x 2 /2500−0.17067=1 x 2 /2500=1.17067 x 2 =2926.675 x= 2926.675 ​ ≈54.1 (x,y) = (x, 60) x 2 /2500−(60) 2 /37500=1 x 2 /2500−3600/37500=1 x 2 /2500−0.096=1 x 2 /2500=1.096 x 2 =2740 x= 2740 ​ ≈52.35