Conic Sections Session 3: Hyperbola
|
|
- Francis Lucas
- 6 years ago
- Views:
Transcription
1 Conic Sections Session 3: Hyperbola Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
2 Problem Recall that an ellipse is defined as the locus of points P such that PF 1 + PF 2 = k. What happens if k F 1 F 2? 2 What about the locus of P where PF 1 PF 2 = k? (Compare the value of k with F 1 F 2.) Q1) The locus is the segment F 1 F 2 when k = F 1 F 2. This can be considered a degenerate ellipse. There are no solutions for smaller values of k. (Triangle inequality.) Q2) If k = F 1 F 2, the locus is a ray from F 2 in the direction of F 1 F 2. If k = 0, it is the line equidistant from F 1 and F 2. If k > F 1 F 2, no solutions since PF 1 PF 2 F 1 F 2. In GeoGebra, create F 1, F 2 such that F 1 F 2 = 2 and slider 5 k 5. The equations PF 2 = r and PF 1 = r + k independently define a pair of circles whose intersections give feasible points for P. To visualize the locus, create slider 0 r 20 and use the locus tool. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
3 Problem 3.2 (Geometric = Cartesian) Derive the Cartesian form of PF 1 PF 2 = 2a where 2a < F 1 F 2. Let F 1 = (c, 0) and F 2 = ( c, 0) with c > 0, c < a < c. (x c) 2 + y 2 (x + c) 2 + y 2 = 2a ( ) 2 = (x c) 2 + y 2 = 2a + (x + c) 2 + y 2 = 4a 2 + 4a (x + c) 2 + y 2 + (x + c) 2 + y 2 = 4xc 4a 2 = 4a (x + c) 2 + y 2 = ( cx + a 2) 2 = a 2 (x + c) 2 + a 2 y 2 = (c 2 a 2 )x 2 a 2 y 2 = a 2 (c 2 a 2 ) = x 2 a 2 y 2 c 2 a 2 = 1. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
4 Definition 3.3 A hyperbola in standard position with transverse axis on x-axis is defined by x 2 a 2 y 2 b 2 = 1. Vertices: A = (a, 0), B = ( a, 0). F2 B A F1 AB is the transverse or major axis. Foci are at (±c, 0) with c = a 2 + b 2. Asymptotes: y = ± b a x. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
5 Definition 3.4 A hyperbola in standard position with transverse axis on y-axis is defined by y 2 b 2 x 2 a 2 = 1. F1 C D F2 Vertices: C = (0, b), D = (0, b). CD is the transverse or major axis. Foci are at (0, ±c) with c = a 2 + b 2. Asymptotes: y = ± b a x. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
6 Definition 3.5 (Parametric Form) The parametric form of the hyperbola x2 y 2 a 2 b 2 = 1 is given by x = ±a cosh t and y = b sinh t, t R. cosh t = et + e t 1 and so a cosh t a. 2 sinh t = et e t and takes on all real values. 2 cosh 2 t sinh 2 t = 1. The gradient of the tangent line to the hyperbola at (x, y) where x > 0 is given by dy dx = b cosh t a sinh t = b2 x a 2 y. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
7 Definition 3.6 (Alternative Parametric Form) The parametric form of the hyperbola x2 y 2 a 2 b 2 = 1 is given by x = a sec θ and y = b tan θ, 0 θ < 2π, θ π 2, 3π 2. Use GeoGebra to plot the locus of intersections of the lines given by the above parametrization and note the discontinuities. Do the same for the parametrization by hyperbolic functions. The gradient of the tangent line to the hyperbola at (x, y) is given by dy dx = b sec t a tan t = b2 x a 2 y. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
8 Problem 3.7 (Cartesian = Geometric ) Let x2 = 1 be a hyperbola and let F b 2 1 be the right focus. Using parametrization or otherwise, show that for an arbitrary point P = (x 0, y 0 ), with x 0 > 0, PF 1 = cx 0 a 2. a PF 1 = (a cosh t a 2 + b 2 ) 2 + b 2 sinh 2 t = a 2 cosh 2 t 2a(cosh t) a 2 + b 2 + a 2 + b 2 + b 2 (cosh 2 t 1) a 2 y 2 = ( a 2 + b 2 cosh t a = c cosh t a = cx 0 a 2 a. ) 2 Similarly PF 2 = c cosh t + a = cx 0+a 2 a. Consequently, PF 2 PF 1 = 2a, if P has positive x-coordinate. For general P, PF1 PF 2 = 2a, a > 0. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
9 Exploring eccentricity Problem 3.8 Let F = (0, 0) and directrix d be the line x = k, with k > 0. If P satisfies PF = e Pd for e 0, find the possible solutions for P that lie on the x-axis. If P = (x, 0), the equation is simply x = e x k. Case: x > k > 0, i.e. x = e(x k) = x = ek e 1. In this case, we see that e > 1. Case: 0 < x < k, i.e. x = e(k x) = x = ek e+1. In this case, there are no additional constraints on e. Case: x = 0, e = 0 (degenerate case). Case: x < 0, i.e. x = e(k x) = x = ek e 1. In this case, we require e < 1. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
10 Definition of Hyperbola Definition 3.9 (Directrix-Eccentricity-Focus) A hyperbola is the locus of points whose distance from the focus F (focus) and { the directrix d are } in a constant ratio greater than 1, i.e. P : PF Pd = e, e > 1. d D P A F Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
11 Varying eccentricity and directrix: Repeat Exercise Exercise 3.10 In GeoGebra, create sliders with variables 0 e 5, 5 k 20 and 0 m 100. Create the directrix d : x = k and the focus F at the origin. Since PF = e Pd, we shall let m be the value of Pd, so P lies on one of the lines x = k ± m. On the other hand, P lies on a circle of radius em about F. (Use circle tool.) Plot the (four possible) intersections to find positions of P. Use the locus tool to draw the conic. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
12 Linking the definitions Definition 3.11 (D-E-F = Cartesian) Let F = (c, 0), c > 0 and directrix d be the line x = c e 2. The equation PF Pd = e for e > 1 is equivalent to e 2 x 2 e 2 y 2 c 2 c 2 (e 2 1) = 1 Thus a = c e and b = (e 2 1)a. Directrix becomes x = a e. e = 1 + b2 a 2 c = ae = a 2 + b 2. (So the focus defined in the G and D-E-F definitions coincide.) By symmetry F 2 = ( c, 0) and x = a e also qualify as focus and directrix. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
13 Linking the definitions Problem 3.12 (D-E-F = Cartesian) Let F = (0, c), c > 0 and directrix d be the line y = k. Show that equation PF Pd = e for e > 1 defines a hyperbola and determine the value of e, c and k for a hyperbola in standard position with transverse axis on y-axis. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
14 Polar form with respect to the focus Problem 3.13 Let F = (0, 0) be a focus and directrix d given by x = k, k > 0. Show that the curve with locus defined by {P : PF = e Pd } has the polar form. r = ek 1 + e cos θ. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
15 Reflection Property Problem 3.14 Show that light coming from F 1 is reflected in a hyperbolic mirror in such a way that it appears to have come from F 2. P F 2 F 1 Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
16 Problem 3.15 (Apollonius) Let T be a point on a hyperbola in standard position with transverse axis on x-axis. The tangent line to the hyperbola at T cuts the asymptotes at P and Q. Show that T is the midpoint of PQ. If O is the origin, show that OQ OP is a constant and hence the triangle OPQ has constant area. Toh Pee Choon (NIE) Session 3: Hyperbola Oct / 16
Conic Sections Session 2: Ellipse
Conic Sections Session 2: Ellipse Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 2: Ellipse Oct 2017 1 / 24 Introduction Problem 2.1 Let A, F 1 and F 2 be three points that form a triangle F 2
More informationThe Distance Formula. The Midpoint Formula
Math 120 Intermediate Algebra Sec 9.1: Distance Midpoint Formulas The Distance Formula The distance between two points P 1 = (x 1, y 1 ) P 2 = (x 1, y 1 ), denoted by d(p 1, P 2 ), is d(p 1, P 2 ) = (x
More informationDistance and Midpoint Formula 7.1
Distance and Midpoint Formula 7.1 Distance Formula d ( x - x ) ( y - y ) 1 1 Example 1 Find the distance between the points (4, 4) and (-6, -). Example Find the value of a to make the distance = 10 units
More informationGeometry and Motion, MA 134 Week 1
Geometry and Motion, MA 134 Week 1 Mario J. Micallef Spring, 2007 Warning. These handouts are not intended to be complete lecture notes. They should be supplemented by your own notes and, importantly,
More informationConic Sections. Geometry - Conics ~1~ NJCTL.org. Write the following equations in standard form.
Conic Sections Midpoint and Distance Formula M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -), find M 2. A(5, 7) and B( -2, -), find M 3. A( 2,0)
More informationEdexcel GCE A Level Maths. Further Maths 3 Coordinate Systems
Edecel GCE A Level Maths Further Maths 3 Coordinate Sstems Edited b: K V Kumaran kumarmaths.weebl.com 1 kumarmaths.weebl.com kumarmaths.weebl.com 3 kumarmaths.weebl.com 4 kumarmaths.weebl.com 5 1. An ellipse
More information3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A
Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(4, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -9), find M 3. A( 2,0)
More informationQUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)
QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents
More informationJanuary 21, 2018 Math 9. Geometry. The method of coordinates (continued). Ellipse. Hyperbola. Parabola.
January 21, 2018 Math 9 Ellipse Geometry The method of coordinates (continued) Ellipse Hyperbola Parabola Definition An ellipse is a locus of points, such that the sum of the distances from point on the
More informationIntroduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves
Introduction to Computer Graphics (Lecture No 07) Ellipse and Other Curves 7.1 Ellipse An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r1 and r from two fixed
More informationCircles. Example 2: Write an equation for a circle if the enpoints of a diameter are at ( 4,5) and (6, 3).
Conics Unit Ch. 8 Circles Equations of Circles The equation of a circle with center ( hk, ) and radius r units is ( x h) ( y k) r. Example 1: Write an equation of circle with center (8, 3) and radius 6.
More informationChapter 10: Conic Sections; Polar Coordinates; Parametric Equations
Chapter 10: Conic Sections; Polar Coordinates; Parametric Equations Section 10.1 Geometry of Parabola, Ellipse, Hyperbola a. Geometric Definition b. Parabola c. Ellipse d. Hyperbola e. Translations f.
More information3. A( 2,0) and B(6, -2), find M 4. A( 3, 7) and M(4,-3), find B. 5. M(4, -9) and B( -10, 11) find A 6. B(4, 8) and M(-2, 5), find A
Midpoint and Distance Formula Class Work M is the midpoint of A and B. Use the given information to find the missing point. 1. A(, 2) and B(3, -8), find M 2. A(5, 7) and B( -2, -), find M (3. 5, 3) (1.
More informationPARAMETRIC EQUATIONS AND POLAR COORDINATES
10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES 10.5 Conic Sections In this section, we will learn: How to derive standard equations for conic sections. CONIC SECTIONS
More informationMATH10000 Mathematical Workshop Project 2 Part 1 Conic Sections
MATH10000 Mathematical Workshop Project 2 Part 1 Conic Sections The aim of this project is to introduce you to an area of geometry known as the theory of conic sections, which is one of the most famous
More informationHonors Precalculus Chapter 8 Summary Conic Sections- Parabola
Honors Precalculus Chapter 8 Summary Conic Sections- Parabola Definition: Focal length: y- axis P(x, y) Focal chord: focus Vertex x-axis directrix Focal width/ Latus Rectum: Derivation of equation of parabola:
More informationLecture 17. Implicit differentiation. Making y the subject: If xy =1,y= x 1 & dy. changed to the subject of y. Note: Example 1.
Implicit differentiation. Lecture 17 Making y the subject: If xy 1,y x 1 & dy dx x 2. But xy y 2 1 is harder to be changed to the subject of y. Note: d dx (f(y)) f (y) dy dx Example 1. Find dy dx given
More information6675/01 Edexcel GCE Pure Mathematics P5 Further Mathematics FP2 Advanced/Advanced Subsidiary
6675/1 Edecel GCE Pure Mathematics P5 Further Mathematics FP Advanced/Advanced Subsidiary Monday June 5 Morning Time: 1 hour 3 minutes 1 1. (a) Find d. (1 4 ) (b) Find, to 3 decimal places, the value of.3
More informationAnalytic Geometry MAT 1035
Analytic Geometry MAT 035 5.09.04 WEEKLY PROGRAM - The first week of the semester, we will introduce the course and given a brief outline. We continue with vectors in R n and some operations including
More informationDAY 139 EQUATION OF A HYPERBOLA
DAY 139 EQUATION OF A HYPERBOLA INTRODUCTION In our prior conic sections lessons, we discussed in detail the two conic sections, the parabola, and the ellipse. The hyperbola is another conic section we
More informationMath Conic Sections
Math 114 - Conic Sections Peter A. Perry University of Kentucky April 13, 2017 Bill of Fare Why Conic Sections? Parabolas Ellipses Hyperbolas Shifted Conics Goals of This Lecture By the end of this lecture,
More informationAnalytic Geometry MAT 1035
Analytic Geometry MAT 035 5.09.04 WEEKLY PROGRAM - The first week of the semester, we will introduce the course and given a brief outline. We continue with vectors in R n and some operations including
More informationCalculus III. George Voutsadakis 1. LSSU Math 251. Lake Superior State University. 1 Mathematics and Computer Science
Calculus III George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 251 George Voutsadakis (LSSU) Calculus III January 2016 1 / 76 Outline 1 Parametric Equations,
More informationMath 190 (Calculus II) Final Review
Math 90 (Calculus II) Final Review. Sketch the region enclosed by the given curves and find the area of the region. a. y = 7 x, y = x + 4 b. y = cos ( πx ), y = x. Use the specified method to find the
More informationPRACTICE PAPER 6 SOLUTIONS
PRACTICE PAPER 6 SOLUTIONS SECTION A I.. Find the value of k if the points (, ) and (k, 3) are conjugate points with respect to the circle + y 5 + 8y + 6. Sol. Equation of the circle is + y 5 + 8y + 6
More informationFrom the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot. Harish Chandra Rajpoot Rajpoot, HCR. Winter February 24, 2015
From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Winter February 24, 2015 Mathematical Analysis of Elliptical Path in the Annular Region Between Two Circles, Smaller Inside the Bigger One
More informationFurther Pure Mathematics 3 GCE Further Mathematics GCE Pure Mathematics and Further Mathematics (Additional) A2 optional unit
Unit FP3 Further Pure Mathematics 3 GCE Further Mathematics GCE Pure Mathematics and Further Mathematics (Additional) A optional unit FP3.1 Unit description Further matrix algebra; vectors, hyperbolic
More informationRevision Checklist. Unit FP3: Further Pure Mathematics 3. Assessment information
Revision Checklist Unit FP3: Further Pure Mathematics 3 Unit description Further matrix algebra; vectors, hyperbolic functions; differentiation; integration, further coordinate systems Assessment information
More informationTime : 3 hours 02 - Mathematics - July 2006 Marks : 100 Pg - 1 Instructions : S E CT I O N - A
Time : 3 hours 0 Mathematics July 006 Marks : 00 Pg Instructions :. Answer all questions.. Write your answers according to the instructions given below with the questions. 3. Begin each section on a new
More informationSenior Math Circles February 11, 2009 Conics II
1 University of Waterloo Faculty of Mathematics Centre for Education in Mathematics and Computing Senior Math Circles February 11, 2009 Conics II Locus Problems The word locus is sometimes synonymous with
More informationSenior Math Circles February 18, 2009 Conics III
University of Waterloo Faculty of Mathematics Senior Math Circles February 18, 2009 Conics III Centre for Education in Mathematics and Computing Eccentricity of Conics Fix a point F called the focus, a
More informationConic Sections in Polar Coordinates
Conic Sections in Polar Coordinates MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction We have develop the familiar formulas for the parabola, ellipse, and hyperbola
More informationMixed exercise 3. x y. cosh t sinh t 1 Substituting the values for cosh t and sinht in the equation for the hyperbola H. = θ =
Mixed exercise x x a Parametric equations: cosθ and sinθ 9 cos θ + sin θ Substituting the values for cos θ and sinθ in the equation for ellipse E gives the Cartesian equation: + 9 b Comparing with the
More informationIIT JEE Maths Paper 2
IIT JEE - 009 Maths Paper A. Question paper format: 1. The question paper consists of 4 sections.. Section I contains 4 multiple choice questions. Each question has 4 choices (A), (B), (C) and (D) for
More informationGrade 11/12 Math Circles Conics & Applications The Mathematics of Orbits Dr. Shahla Aliakbari November 18, 2015
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 11/12 Math Circles Conics & Applications The Mathematics of Orbits Dr. Shahla Aliakbari November
More informationy 1 x 1 ) 2 + (y 2 ) 2 A circle is a set of points P in a plane that are equidistant from a fixed point, called the center.
Ch 12. Conic Sections Circles, Parabolas, Ellipses & Hyperbolas The formulas for the conic sections are derived by using the distance formula, which was derived from the Pythagorean Theorem. If you know
More informationALGEBRA 2 X. Final Exam. Review Packet
ALGEBRA X Final Exam Review Packet Multiple Choice Match: 1) x + y = r a) equation of a line ) x = 5y 4y+ b) equation of a hyperbola ) 4) x y + = 1 64 9 c) equation of a parabola x y = 1 4 49 d) equation
More informationMath 259 Winter Solutions to Homework # We will substitute for x and y in the linear equation and then solve for r. x + y = 9.
Math 59 Winter 9 Solutions to Homework Problems from Pages 5-5 (Section 9.) 18. We will substitute for x and y in the linear equation and then solve for r. x + y = 9 r cos(θ) + r sin(θ) = 9 r (cos(θ) +
More informationConic section. Ans: c. Ans: a. Ans: c. Episode:43 Faculty: Prof. A. NAGARAJ. 1. A circle
Episode:43 Faculty: Prof. A. NAGARAJ Conic section 1. A circle gx fy c 0 is said to be imaginary circle if a) g + f = c b) g + f > c c) g + f < c d) g = f. If (1,-3) is the centre of the circle x y ax
More informationExtra FP3 past paper - A
Mark schemes for these "Extra FP3" papers at https://mathsmartinthomas.files.wordpress.com/04//extra_fp3_markscheme.pdf Extra FP3 past paper - A More FP3 practice papers, with mark schemes, compiled from
More informationTWO THEOREMS ON THE FOCUS-SHARING ELLIPSES: A THREE-DIMENSIONAL VIEW
TWO THEOREMS ON THE FOCUS-SHARING ELLIPSES: A THREE-DIMENSIONAL VIEW ILYA I. BOGDANOV Abstract. Consider three ellipses each two of which share a common focus. The radical axes of the pairs of these ellipses
More informationDevice Constructions with Hyperbolas
lfonso Croeze 1 William Kelly 1 William Smith 2 1 Department of Mathematics Louisiana State University aton Rouge, L 2 Department of Mathematics University of Mississippi Oxford, MS July 8, 2011 Hyperbola
More informationTARGET : JEE 2013 SCORE. JEE (Advanced) Home Assignment # 03. Kota Chandigarh Ahmedabad
TARGT : J 01 SCOR J (Advanced) Home Assignment # 0 Kota Chandigarh Ahmedabad J-Mathematics HOM ASSIGNMNT # 0 STRAIGHT OBJCTIV TYP 1. If x + y = 0 is a tangent at the vertex of a parabola and x + y 7 =
More informationRotation of Axes. By: OpenStaxCollege
Rotation of Axes By: OpenStaxCollege As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions,
More informationObjective Mathematics
. A tangent to the ellipse is intersected by a b the tangents at the etremities of the major ais at 'P' and 'Q' circle on PQ as diameter always passes through : (a) one fied point two fied points (c) four
More informationOHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1
OHSx XM521 Multivariable Differential Calculus: Homework Solutions 13.1 (37) If a bug walks on the sphere x 2 + y 2 + z 2 + 2x 2y 4z 3 = 0 how close and how far can it get from the origin? Solution: Complete
More information9.1 Circles and Parabolas. Copyright Cengage Learning. All rights reserved.
9.1 Circles and Parabolas Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize a conic as the intersection of a plane and a double-napped cone. Write equations of circles in
More informationKEMATH1 Calculus for Chemistry and Biochemistry Students. Francis Joseph H. Campeña, De La Salle University Manila
KEMATH1 Calculus for Chemistry and Biochemistry Students Francis Joseph H Campeña, De La Salle University Manila February 9, 2015 Contents 1 Conic Sections 2 11 A review of the coordinate system 2 12 Conic
More informationMA 162 FINAL EXAM PRACTICE PROBLEMS Spring Find the angle between the vectors v = 2i + 2j + k and w = 2i + 2j k. C.
MA 6 FINAL EXAM PRACTICE PROBLEMS Spring. Find the angle between the vectors v = i + j + k and w = i + j k. cos 8 cos 5 cos D. cos 7 E. cos. Find a such that u = i j + ak and v = i + j + k are perpendicular.
More informationChapter 1 Analytic geometry in the plane
3110 General Mathematics 1 31 10 General Mathematics For the students from Pharmaceutical Faculty 1/004 Instructor: Dr Wattana Toutip (ดร.ว ฒนา เถาว ท พย ) Chapter 1 Analytic geometry in the plane Overview:
More informationPractice Assessment Task SET 3
PRACTICE ASSESSMENT TASK 3 655 Practice Assessment Task SET 3 Solve m - 5m + 6 $ 0 0 Find the locus of point P that moves so that it is equidistant from the points A^-3, h and B ^57, h 3 Write x = 4t,
More information3.4 Conic sections. Such type of curves are called conics, because they arise from different slices through a cone
3.4 Conic sections Next we consider the objects resulting from ax 2 + bxy + cy 2 + + ey + f = 0. Such type of curves are called conics, because they arise from different slices through a cone Circles belong
More informationTHE NCUK INTERNATIONAL FOUNDATION YEAR (IFY) Further Mathematics
IFYFM00 Further Maths THE NCUK INTERNATIONAL FOUNDATION YEAR (IFY) Further Mathematics Examination Session Summer 009 Time Allowed hours 0 minutes (Including 0 minutes reading time) INSTRUCTIONS TO STUDENTS
More informationParametric Equations and Polar Coordinates
Parametric Equations and Polar Coordinates Parametrizations of Plane Curves In previous chapters, we have studied curves as the graphs of functions or equations involving the two variables x and y. Another
More informationMathematics Extension 2
Northern Beaches Secondary College Manly Selective Campus 010 HIGHER SCHOOL CERTIFICATE TRIAL EXAMINATION Mathematics Extension General Instructions Reading time 5 minutes Working time 3 hours Write using
More informationStandardized Test Practice
Standardized Test Practice. A store uses a matrix to show their inventory of jeans by waist size (in inches) and style of leg. What is a 3? A a. straight boot cut flared tapered 3 3 3 3 3 3 7 3 3 9 b.
More informationON THE INTRODUCTION OF THE NOTION OF HYPEEBOLIC FUNCTIONS.*
1895] THE NOTION OF HYPERBOLIC FUNCTIONS. 155 ON THE INTRODUCTION OF THE NOTION OF HYPEEBOLIC FUNCTIONS.* BY PROFESSOR M. W. HASKELL. THE difficulties in the way of a satisfactory geometrical deduction
More informationy mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent
Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()
More informationMAT1035 Analytic Geometry
MAT1035 Analytic Geometry Lecture Notes R.A. Sabri Kaan Gürbüzer Dokuz Eylül University 2016 2 Contents 1 Review of Trigonometry 5 2 Polar Coordinates 7 3 Vectors in R n 9 3.1 Located Vectors..............................................
More informationb = 2, c = 3, we get x = 0.3 for the positive root. Ans. (D) x 2-2x - 8 < 0, or (x - 4)(x + 2) < 0, Therefore -2 < x < 4 Ans. (C)
SAT II - Math Level 2 Test #02 Solution 1. The positive zero of y = x 2 + 2x is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E) 2.2 ± Using Quadratic formula, x =, with a = 1,
More informationchapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?
chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "
More informationSpacecraft Dynamics and Control
Spacecraft Dynamics and Control Matthew M. Peet Arizona State University Lecture 5: Hyperbolic Orbits Introduction In this Lecture, you will learn: Hyperbolic orbits Hyperbolic Anomaly Kepler s Equation,
More informationPrecalculus Conic Sections Unit 6. Parabolas. Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix
PICTURE: Parabolas Name Hr Label the parts: Focus Vertex Axis of symmetry Focal Diameter Directrix Using what you know about transformations, label the purpose of each constant: y a x h 2 k It is common
More informationCSSA Trial HSC Examination
CSSA Trial HSC Examination. (a) Mathematics Extension 2 2002 The diagram shows the graph of y = f(x) where f(x) = x2 x 2 +. (i) Find the equation of the asymptote L. (ii) On separate diagrams sketch the
More informationThings you should have learned in Calculus II
Things you should have learned in Calculus II 1 Vectors Given vectors v = v 1, v 2, v 3, u = u 1, u 2, u 3 1.1 Common Operations Operations Notation How is it calculated Other Notation Dot Product v u
More informationCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 2012, Brooks/Cole
More information2. Determine the domain of the function. Verify your result with a graph. f(x) = 25 x 2
29 April PreCalculus Final Review 1. Find the slope and y-intercept (if possible) of the equation of the line. Sketch the line: y = 3x + 13 2. Determine the domain of the function. Verify your result with
More informationMathematics Precalculus: Academic Unit 7: Conics
Understandings Questions Knowledge Vocabulary Skills Conics are models of real-life situations. Conics have many reflective properties that are used in every day situations Conics work can be simplified
More informationSchool of Distance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. B Sc Mathematics. (2011 Admission Onwards) IV Semester.
School of Dtance Education UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc Mathematics 0 Admsion Onwards IV Semester Core Course CALCULUS AND ANALYTIC GEOMETRY QUESTION BANK The natural logarithm
More informationPre Calculus Gary Community School Corporation Unit Planning Map
UNIT/TIME FRAME STANDARDS Functions and Graphs (6 weeks) PC.F.1: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities,
More informationCONIC SECTIONS TEST FRIDAY, JANUARY 5 TH
CONIC SECTIONS TEST FRIDAY, JANUARY 5 TH DAY 1 - CLASSIFYING CONICS 4 Conics Parabola Circle Ellipse Hyperbola DAY 1 - CLASSIFYING CONICS GRAPHICALLY Parabola Ellipse Circle Hyperbola DAY 1 - CLASSIFYING
More informationECM Calculus and Geometry. Revision Notes
ECM1702 - Calculus and Geometry Revision Notes Joshua Byrne Autumn 2011 Contents 1 The Real Numbers 1 1.1 Notation.................................................. 1 1.2 Set Notation...............................................
More information1. The positive zero of y = x 2 + 2x 3/5 is, to the nearest tenth, equal to
SAT II - Math Level Test #0 Solution SAT II - Math Level Test No. 1. The positive zero of y = x + x 3/5 is, to the nearest tenth, equal to (A) 0.8 (B) 0.7 + 1.1i (C) 0.7 (D) 0.3 (E). 3 b b 4ac Using Quadratic
More informationPaper Reference. Paper Reference(s) 6669/01 Edexcel GCE Further Pure Mathematics FP3 Advanced/Advanced Subsidiary
Centre No. Candidate No. Surname Signature Paper Reference(s) 6669/01 Edexcel GCE Further Pure Mathematics FP3 Advanced/Advanced Subsidiary Monday 28 June 2010 Afternoon Time: 1 hour 30 minutes Materials
More informationPortable Assisted Study Sequence ALGEBRA IIB
SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The second half of
More informationA plane in which each point is identified with a ordered pair of real numbers (x,y) is called a coordinate (or Cartesian) plane.
Coordinate Geometry Rene Descartes, considered the father of modern philosophy (Cogito ergo sum), also had a great influence on mathematics. He and Fermat corresponded regularly and as a result of their
More information10.1 Review of Parametric Equations
10.1 Review of Parametric Equations Recall that often, instead of representing a curve using just x and y (called a Cartesian equation), it is more convenient to define x and y using parametric equations
More informationSKILL BUILDER TEN. Graphs of Linear Equations with Two Variables. If x = 2 then y = = = 7 and (2, 7) is a solution.
SKILL BUILDER TEN Graphs of Linear Equations with Two Variables A first degree equation is called a linear equation, since its graph is a straight line. In a linear equation, each term is a constant or
More information10550 PRACTICE FINAL EXAM SOLUTIONS. x 2 4. x 2 x 2 5x +6 = lim x +2. x 2 x 3 = 4 1 = 4.
55 PRACTICE FINAL EXAM SOLUTIONS. First notice that x 2 4 x 2x + 2 x 2 5x +6 x 2x. This function is undefined at x 2. Since, in the it as x 2, we only care about what happens near x 2 an for x less than
More informationH2 MATHS SET D PAPER 1
H Maths Set D Paper H MATHS Exam papers with worked solutions SET D PAPER Compiled by THE MATHS CAFE P a g e b The curve y ax c x 3 points, and, H Maths Set D Paper has a stationary point at x 3. It also
More information1 is equal to. 1 (B) a. (C) a (B) (D) 4. (C) P lies inside both C & E (D) P lies inside C but outside E. (B) 1 (D) 1
Single Correct Q. Two mutuall perpendicular tangents of the parabola = a meet the ais in P and P. If S is the focus of the parabola then l a (SP ) is equal to (SP ) l (B) a (C) a Q. ABCD and EFGC are squares
More informationPage 1
Pacing Chart Unit Week Day CCSS Standards Objective I Can Statements 121 CCSS.MATH.CONTENT.HSG.C.A.1 Prove that all circles are similar. Prove that all circles are similar. I can prove that all circles
More informationA. Correct! These are the corresponding rectangular coordinates.
Precalculus - Problem Drill 20: Polar Coordinates No. 1 of 10 1. Find the rectangular coordinates given the point (0, π) in polar (A) (0, 0) (B) (2, 0) (C) (0, 2) (D) (2, 2) (E) (0, -2) A. Correct! These
More informationTopic 2 [312 marks] The rectangle ABCD is inscribed in a circle. Sides [AD] and [AB] have lengths
Topic 2 [312 marks] 1 The rectangle ABCD is inscribed in a circle Sides [AD] and [AB] have lengths [12 marks] 3 cm and (\9\) cm respectively E is a point on side [AB] such that AE is 3 cm Side [DE] is
More informationCIRCLES: #1. What is an equation of the circle at the origin and radius 12?
1 Pre-AP Algebra II Chapter 10 Test Review Standards/Goals: E.3.a.: I can identify conic sections (parabola, circle, ellipse, hyperbola) from their equations in standard form. E.3.b.: I can graph circles
More informationCenterville High School Curriculum Mapping Algebra II 1 st Nine Weeks
Centerville High School Curriculum Mapping Algebra II 1 st Nine Weeks Chapter/ Lesson Common Core Standard(s) 1-1 SMP1 1. How do you use a number line to graph and order real numbers? 2. How do you identify
More informationMath 234. What you should know on day one. August 28, You should be able to use general principles like. x = cos t, y = sin t, 0 t π.
Math 234 What you should know on day one August 28, 2001 1 You should be able to use general principles like Length = ds, Area = da, Volume = dv For example the length of the semi circle x = cos t, y =
More informationExam 4 SCORE. MA 114 Exam 4 Spring Section and/or TA:
Exam 4 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may
More informationby Abhijit Kumar Jha
SET I. If the locus of the point of intersection of perpendicular tangents to the ellipse x a circle with centre at (0, 0), then the radius of the circle would e a + a /a ( a ). There are exactl two points
More informationFunctions, Graphs, Equations and Inequalities
CAEM DPP Learning Outcomes per Module Module Functions, Graphs, Equations and Inequalities Learning Outcomes 1. Functions, inverse functions and composite functions 1.1. concepts of function, domain and
More information5 t + t2 4. (ii) f(x) = ln(x 2 1). (iii) f(x) = e 2x 2e x + 3 4
Study Guide for Final Exam 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its expression to be well-defined. Some examples of the conditions are: What
More information8.6 Translate and Classify Conic Sections
8.6 Translate and Classify Conic Sections Where are the symmetric lines of conic sections? What is the general 2 nd degree equation for any conic? What information can the discriminant tell you about a
More informationAEA 2007 Extended Solutions
AEA 7 Extended Solutions These extended solutions for Advanced Extension Awards in Mathematics are intended to supplement the original mark schemes, which are available on the Edexcel website.. (a The
More informationM GENERAL MATHEMATICS -2- Dr. Tariq A. AlFadhel 1 Solution of the First Mid-Term Exam First semester H
M 4 - GENERAL MATHEMATICS -- Dr. Tariq A. AlFadhel Solution of the First Mid-Term Exam First semester 435-436 H Q. Let A ( ) 4 and B 3 3 Compute (if possible) : AB and BA ( ) 4 AB 3 3 ( ) ( ) ++ 4+4+ 4
More informationGLOBAL TALENT SEARCH EXAMINATIONS (GTSE) CLASS -XI
GLOBAL TALENT SEARCH EXAMINATIONS (GTSE) Date: rd November 008 CLASS -XI MATHEMATICS Max Marks: 80 Time: :0 to :5 a.m. General Instructions: (Read Instructions carefully). All questions are compulsory.
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE TRIAL EXAMINATION Mathematics Extension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Approved scientific calculators and templates
More informationSOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253
SOLUTIONS TO HOMEWORK ASSIGNMENT #, Math 5. Find the equation of a sphere if one of its diameters has end points (, 0, 5) and (5, 4, 7). The length of the diameter is (5 ) + ( 4 0) + (7 5) = =, so the
More informationa Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
More information4. Alexandrian mathematics after Euclid II. Apollonius of Perga
4. Alexandrian mathematics after Euclid II Due to the length of this unit, it has been split into three parts. Apollonius of Perga If one initiates a Google search of the Internet for the name Apollonius,
More informationThe details of the derivation of the equations of conics are com-
Part 6 Conic sections Introduction Consider the double cone shown in the diagram, joined at the verte. These cones are right circular cones in the sense that slicing the double cones with planes at right-angles
More information