DOCSLIB.ORG

Conic Sections : Case (I) : When the Focus Lies on the Directrix

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Conic Sections : Case (I) : When the Focus Lies on the Directrix 1. CONIC SECTIONS : CASE (I) : WHEN THE FOCUS LIES ON THE DIRECTRIX. (i) Vertex is (0, 0) A conic section, or conic is the locus of a point In this case D = abc + 2fgh - af2 - bg2 - ch2 = (ii) focus is (a, 0) which moves in a plane so that its distance 0 & the general equation of a conic (iii) Axis is y = 0 from a fixed point is in a constant ratio to its represents a pair of straight lines if : (iv) Directrix is x + a = 0 perpendicular distance from a fixed straight line e > 1 the line will be real & distinct ➢ The fixed point is called the FOCUS. intersecting at S. ➢ The fixed straight line is called the FOCAL DISTANCE : The distance of a point on E = 1 the line will coincident. DIRECTRIX. the parabola from the focus is called the e < 1 the lines will be imaginary. ➢ The constant ratio is called the FOCAL DISTANCE OF THE POINT. ECCENTRICITY denoted by e. ➢ The line passing through the focus & FOCAL CHOD : perpendicular to the distance called A chord of the parabola, which passes through CASE (II) : WHEN THE FOCUS DOES NOT the axis the focus is called a FOCAL CHORD. LIE ON DIRECTRIX. ➢ A point of intersection of a conic with A parabola an ellipse a hyperbola its axis is called a VERTEX. DOUBLE ORDINATE : A chord of the parabola rectangular hyperbola perpendicular to the axis of the symmetry is C = 1 ; D ≠ 0, 0 < e < 1 ; D ≠ 0; e > 1 ; D ≠ 0; e called a DOUBLE ORDINATE. > 1 ; D ≠ 0 h2 = ab, h2 < ab, h2 > ab, h2 > ab ; a 2. GENERAL EQUATION OF A CONIC : + b = 0 LATUS RECTUM : FOCAL DIRECTRIX PROPERTY : A double ordinate passing through the focus The general equation of a conic with focus 4. PARABOLA : or a focal chord perpendicular to the axis of (p, q) % directrix lx + my + n = 0 is : A parabola is the locus of a point which parabola is called the LATUS RECTUM. For (l2 + m2)[(x - p)2 + (y - q)2] = e2(lx + my + n)2 = moves in a plane, such that its distance from Y2 = 4ax. ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 a fixed point (focus) is equal to its Length of the latus rectum = 4a. perpendicular distance from a fixed straight Ends of the latus rectum are L(a,2a) & L’(a,-2a). 3. DISTINGUISHING BETWEEN THE CONIC : line (directrix). Note that : The nature of the conic section depends upon Standard equation of the parabola is y2 = (i) perpendicular distance from focus on the position of the focus S w.r.t. The directrix & 4ax. For this parabola : directrix = half the latus rectum. also upto the value of the eccentricity e. Two different cases arise. (ii) Vertex is middle point of the focus & the (c) If the normals of the parabola y2 = 4ax at the point of intersection of directrix & axis. Note : If chord joining t t & t t pass a 1 2 3 4 points t & t intersect again on the parabola at (iii) Two parabolas are laid to be equal if they through point (c,0) on axis, then t t = t t = -c/a. 1 2 1 2 3 4, the point ‘t ’ then t t = 2; t = -(t + t ) and the have the same latus rectum. Four standard 3 1 2 3 1 3 line joining t & t passes through a fixed point forms of the parabola are y2 = 4ax ; y2 = - 4ax ; 9. TANGENTS TO THE PARABOLA y2 = 4ax : 1 2 2 2 (-2a, 0). X = 4ay; x = - 4ay (i) yy1 = 2a (x + x1 ) at the point ( x1 , y1 ) ; (ii) y = mx + (m ≠ o) at General Note : 5. POSITION OF A POINT RELATIVE TO A (iii) ty = x + at2 at (at2, 2at). (i) Length of subtangent at any point P(x, y) on PARABOLA : Note : Point of intersection of the tangents at the parabola y2 = 4ax equals twice the abscissa The point ( x , y ) lies outside, on or inside the 1 1 the point t & t is [ at t a(t + t ) ]. of the point P. Note that the subtangent is parabola y2 = 4ax according as the expression 1 2 1 2 1 2 bisected at the vertex. y 2 - 4ax is positive zero or negative. 1 1 10. NORMALS TO THE PARABOLA y2 = 4ax (ii) Length of subnormal is constant for all : points on the parabola & is equal to the semi 6. LINE & A PARABOLA : (i) y - y = latus rectum. The line y = mx + c meets the parabola y2 = 4ax 1 (ii) y = mx - 2m - am3 at (am2 - 2am) (iii) If a family of straight lines can be in two points real, coincident or imaginary (iii) y + tx = 2at + at3 at (at2, 2at). represented by an equation λ2P + λQ + R = 0 according as a ≷ cm ⇒ condition of tangency where λ is a parameter and P, Q, R are linear is, Note : Point of intersection of normals at, t & t 1 3 functions of x and y then the family of lines will are, a(t 2 + t 2 + t 1 2 1 be tangent to the curve Q2 = 4PR. 7. Length of the chord intercepted by the parabola on the line y = mx + c is : 11. THREE VERY IMPORTANT RESULTS : (a) If t1 & t2 are the ends of a focal chord 12. The equation to the pair of tangents which of the parabola y2 = 4ax then t t = -1. 8. PARAMETRIC REPRESENTATION : 1 2 can be drawn from any point (x1, y1) to the Hence the coordinates at the 2 2 The simplest & best form of representing the parabola y = 4ax is given by : SS1 = T where : 2 extremities of a focal chord can be 2 2 coordinates of a point on the parabola is (at , S ≡ y - 4ax ; S1 ≡ y1 - 4ax1 ; T ≡ yy1 - 2a( x + x1). 2at). The equations X = at2 & y = 2at together taken as (at2, 2at ) & represents the parabola y2 = 4ax, t being the (b) If the normals to the parabola y2 = 4ax at the point t , meets the parameter. The equation of a chord joining t1 & 1 parabola again at the point t , then t2 is 2x - (t1 + t2)y + 2 at t1t2 = 0. 2 13.DIRECTOR CIRCLE : The tangent at the point. Locus of the point of intersection of the (iii) A line segment from a point P on the (iii) If the polar of a point P passes through the perpendicular tangents to the parabola y2 = parabola and parallel to the system of parallel point Q, then the polar of Q goes through P. 4ax is called the DIRECTOR CIRCLE. It’s chords is called the ordinate to the diameter (iv) Two straight lines are said to be conjugated equation is x + a = 0 which is parabola’s own bisecting the system of parallel chords and the to each other w.r.t. A parabola when the pole of directrix. chords are called its double ordinate. one lies on the other. 14. CHORD OF CONTACT : (v) Plar of a given point P w.r.t. Any Conic is the 18. IMPORTANT HIGHLIGHTS : Equation to the chord of contact of tangent locus of the harmonic conjugate of P w.r.t. The (a) If the tangents & normal at any point ‘P’ drawn from a point P(x , y ) is yy = 2a(x + x ). two points is which any line though P cuts the 1 1 1 1 of the parabola intersect the axis at T & Remember that the area of the triangle formed conic. G then ST = SG = SP where ‘S’ is the by the tangents from the point (x , y ) & the 1 1 16. CHORD WITH A GIVEN MIDDLE POINT : focus. In other words the tangent and chord of contact is (y 2 - 4ax )3/2 ÷ 2a . Also note 1 1 Equation of the chord of the parabola y2 = 4ax the normal at a point P on the parabola that the chord of contact exists only if the point whose middle point is (x , y ) is are the bisector of the angle between P is not inside. 1 1 This reduced to T = S1 the focal radius SP & the perpendicular 2 Where T ≡ yy1 - 2a( x + x1) & S1 ≡ y1 - 4ax1. from P on the directrix. From this we conclude that all rays emanating from 17. DIAMETER : S will become parallel to the axis of the 15. POLAR & POLE : The locus of the middle points of a system of parabola after reflection. (i) Equation of the Polar of the point P(x , y ) 1 1 parallel chords of a Parabola is called a (b) The point of a tangent to a parabola w.r.t. The parabola y2 = 4ax is yy = 2a(x + x ) 1 1 DIAMETER. Equation to the diameter of a cut off between the directrix & the (ii) The pole of the line lx + my + n = 0 w.r.t. The parabola is y = 2a/m, where m = slope of curve subtends a right angle at focus, parabola y2 = 4ax is parabola chords. (c) The tangents at the extremities of a focal chord intersect at right angles on Note : NOTE : the directrix, and hence a circle on any (i) The polar of the form of the parabola is the (i) The tangent at the extremity of a diameter focal chord as diameter touches the directrix. of a parabola is parallel to the system of chords directrix. Also a circle on any focal radii (ii) When the point (x , y ) lies without the 1 1 it bisects. of a point P(at2, 2at) as diameter parabola the equation to its polar is the same (ii) The tangent the ends of any chords of a as the equation to the chord of contact of parabola meet on the diameter which bisects tangents drawn from (x , y ) when (x , y ) is on 1 1 1 1 the chord.
Load more

Recommended publications
  • Automated Study of Isoptic Curves: a New Approach with Geogebra∗
    Automated study of isoptic curves: a new approach with GeoGebra∗ Thierry Dana-Picard1 and Zoltan Kov´acs2 1 Jerusalem College of Technology, Israel [email protected] 2 Private University College of Education of the Diocese of Linz, Austria [email protected] March 7, 2018 Let C be a plane curve. For a given angle θ such that 0 ≤ θ ≤ 180◦, a θ-isoptic curve of C is the geometric locus of points in the plane through which pass a pair of tangents making an angle of θ.If θ = 90◦, the isoptic curve is called the orthoptic curve of C. For example, the orthoptic curves of conic sections are respectively the directrix of a parabola, the director circle of an ellipse, and the director circle of a hyperbola (in this case, its existence depends on the eccentricity of the hyperbola). Orthoptics and θ-isoptics can be studied for other curves than conics, in particular for closed smooth strictly convex curves,as in [1]. An example of the study of isoptics of a non-convex non-smooth curve, namely the isoptics of an astroid are studied in [2] and of Fermat curves in [3]. The Isoptics of an astroid present special properties: • There exist points through which pass 3 tangents to C, and two of them are perpendicular. • The isoptic curve are actually union of disjoint arcs. In Figure 1, the loops correspond to θ = 120◦ and the arcs connecting them correspond to θ = 60◦. Such a situation has been encountered already for isoptics of hyperbolas. These works combine geometrical experimentation with a Dynamical Geometry System (DGS) GeoGebra and algebraic computations with a Computer Algebra System (CAS).
    [Show full text]
  • HYPERBOLA HYPERBOLA a Conic Is Said to Be a Hyperbola If Its Eccentricity Is Greater Than One
    HYPERBOLA HYPERBOLA A Conic is said to be a hyperbola if its eccentricity is greater than one. If S ax22 hxy by 2 2 gx 2 fy c 0 represents a hyperbola then h2 ab 0 and 0 2 2 If S ax2 hxy by 2 gx 2 fy c 0 represents a rectangular hyperbola then h2 ab 0 , 0 and a b 0 x2 y 2 Hyperbola: Equation of the hyperbola in standard form is 1. a2 b2 Hyperbola is symmetric about both the axes. Hyperbola does not passing through the origin and does not meet the Y - axis. Hyperbola curve does not exist between the lines x a and x a . x2 y 2 xx yy x2 y 2 x x y y Notation: S 1 , S1 1 1, S1 1 1, S1 2 1 2 1. a2 b 2 1 a2 b 2 11 a2 b 2 12 a2 b 2 Rectangular hyperbola (or) Equilateral hyperbola : In a hyperbola if the length of the transverse axis (2a) is equal to the length of the conjugate axis (2b) , then the hyperbola is called a rectangular hyperbola or an equilateral hyperbola. The eccentricity of a rectangular hyperbola is 2 . Conjugate hyperbola : The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called the conjugate hyperbola of the given hyperbola. 2 2 1 x y The equation of the hyperbola conjugate to S 0 is S 1 0 . a2 b 2 If e1 and e2 be the eccentricities of the hyperbola and its conjugate hyperbola then 1 1 2 2 1.
    [Show full text]
  • Department of Mathematics Scheme of Studies B.Sc
    BHUPAL NOBLES` UNIVERSITY, UDAIPUR FACULTY OF SCIENCE Department Of Mathematics Scheme of Studies B.Sc. I Year (Annual Scheme) S. No. PAPER NOMENCLATURE COURSE UNIVERSITY INTERNAL MAX. CODE EXAM ASSESSMENT MARKS 1. Paper I Algebra MAT-111 53 22 75 2. Paper II Calculus MAT- 112 53 22 75 3. Paper III Geometry MAT- 113 53 22 75 The marks distribution of internal assessment- 1. Mid Term Examination – 15 marks 2. Attendance – 7 marks B.Sc. II Year (Annual Scheme) S. No. PAPER NOMENCLATURE COURSE UNIVERSITY INTERNAL MAX. CODE EXAM ASSESSMENT MARKS 1. Paper I Advanced Calculus MAT-221 53 22 75 2. Paper II Differential Equation MAT- 222 53 22 75 3. Paper III Mechanics MAT- 223 53 22 75 The marks distribution of internal assessment- 1. Mid Term Examination – 15 marks 2. Attendance – 7 marks B.Sc. III Year (Annual Scheme) S. PAPER NOMENCLATURE COURSE CODE UNIVERSITY INTERNAL MAX. No. EXAM ASSESSMENT MARKS 1. Paper I Real Analysis MAT-331 53 22 75 2. Paper II Advanced Algebra MAT- 332 53 22 75 3. Paper Numerical Analysis MAT-333 (A) 53 22 75 III Mathematical MAT-333 (B) Quantitative Techniques Mathematical MAT-333 (C) Statistics The marks distribution of internal assessment- 1. Mid Term Examination – 15 marks 2. Attendance – 7 marks BHUPAL NOBLES’ UNIVERSITY, UDAIPUR Department of Mathematics Syllabus 2017 -2018 (COMMON FOR THE FACULTIES OF ARTS & SCIENCE) B.A. / B. Sc. FIRST YEAR EXAMINATIONS 2017-2020 MATHEMATICS Theory Papers Name Papers Code Papers Maximum Papers hours/ Marks BA/ week B.Sc. Paper I ALGEBRA MAT-111 3 75 Paper II CALCULUS MAT-112 3 75 Paper III GEOMETRY MAT-113 3 75 Total Marks 225 NOTE: 1.
    [Show full text]
  • Syllabus for Bachelor of Science I-Year (Semester-I) Session 2018
    Syllabus for Bachelor of Science I-Year (Semester-I) Session 2018-2019 Code Description Pd/W Exam CIA ESE Total BSMT111 Algebra 3 3 hrs 20 80 100 BSMT112 Differential Calculus 3 3 hrs 20 80 100 BSMT113 Co-ordinate Geometry of Two and 3 3 hrs 20 80 100 Three Dimensions Total 60 240 300 BSMT111: ALGEBRA Matrix: The characteristic equation of matrix: Eigen Values and Eigen Unit-I Vectors, Diagonalization of matrix, Cayley-Hamilton theorem(Statement 9 and proof),and its use in finding the inverse of a Matrix. Theory of equation: Relation between roots and coefficient of the equation Unit-II Symmetric function of roots. Solution of cubic equation by Cordon’s 9 method and Biquadratic equitation by Ferrari’s method. Infinite series: Convergent series, convergence of geometric series, And Unit-III necessary condition for the convergent series, comparison tests: Cauchy 9 root test. D’Alembert’s Ratio test, Logarithmic test, Raabe’s test, De’ Morgan and Unit-IV Bertrand’s test, Cauchy’s condensation test, Leibnitz’s test of alternative 9 series, Absolute convergent. Unit-V Inequalities and Continued fractions. 9 Suggested Readings: 1. Bhargav and Agarwal:Algebra, Jaipur publishing House, Jaipur. 2. Vashistha and Vashistha : Modern Algebra, Krishna Prakashan, Meerut. 3. Gokhroo, Saini and Tak: Algebra,Navkar prakashan, Ajmer.. 4. M-Ray and H. S. Sharma: A Text book of Higher Algebra, New Delhi, BSMT112: Differential Calculus Unit-I Polar Co-ordinates, Angle between radius vector and the tangents. Angle 9 between curves in polar form, length of polar subs tangent and subnormal, pedal equation of a curve, derivatives of an arc.
    [Show full text]
  • Group B Higher Algebra & Trigonometry
    UNIVERSITY DEPARTMENT OF MATHEMATICS, DSPMU, RANCHI CBCS PATTERN SYLLABUS Semester Mat/ Sem VC1- Analytical Geometry 2D, Trigonometry Instruction: Ten questions will be set. Candidates will be required to answer Seven Questions. Question no. 1 will be Compulsory consisting of 10 short answer type covering entire syllabus uniformly. Each question will be of 2 marks. Out of remaining 9 questions candidates will be required to answer 6 questions selecting at least one from each group. Each question will be of 10 marks. GROUP-A ANALYTICAL GEOMETRY OF TWO DIMENSIONS Change of rectangular axes. Condition for the general equation of second degree to represent parabola, ellipse, hyperbola and reduction into standard forms. Equations of tangent and normal (Using Calculus). Chord of contact, Pole and Polar. Pair of tangents in reference to general equation of conic. Axes, centre, director circle in reference to general equation of conic. Polar equation of conic. 5 Questions GROUP B HIGHER ALGEBRA & TRIGONOMETRY Statement and proof of binomial theorem for any index, exponential and logarithmic series. 1 Question De Moivre's theorem and its applications. Trigonometric and Exponential functions of complex argument and hyperbolic functions. Summation of Trigonometrical series. Factorisation of sin , cos 3 Questions Books Recommended: 1. Analytical Geometry& Vector Analysis - B. K. Kar, Books& Allied Co., Kolkata 2. Analytical Geometry of two dimension - Askwith 3. Coordinate Geometry- SL Loney. 4. Trigonometry- Das and Mukheriee 5. Trigonometry- Dasgupta UNIVERSITY DEPARTMENT OF MATHEMATICS, DSPMU, RANCHI CBCS PATTERN SYLLABUS Semester Mat/ Sem IVC 2- Differential Calculus and Vector Calculus Instruction: Ten questions will be set. Candidates will be required to answer Seven Questions.
    [Show full text]
  • 1 CHAPTER 2 CONIC SECTIONS 2.1 Introduction a Particle Moving Under the Influence of an Inverse Square Force Moves in an Orbit T
    1 CHAPTER 2 CONIC SECTIONS 2.1 Introduction A particle moving under the influence of an inverse square force moves in an orbit that is a conic section; that is to say an ellipse, a parabola or a hyperbola. We shall prove this from dynamical principles in a later chapter. In this chapter we review the geometry of the conic sections. We start off, however, with a brief review (eight equation-packed pages) of the geometry of the straight line. 2.2 The Straight Line It might be thought that there is rather a limited amount that could be written about the geometry of a straight line. We can manage a few equations here, however, (there are 35 in this section on the Straight Line) and shall return for more on the subject in Chapter 4. Most readers will be familiar with the equation y = mx + c 2.2.1 for a straight line. The slope (or gradient) of the line, which is the tangent of the angle that it makes with the x-axis, is m, and the intercept on the y-axis is c. There are various other forms that may be of use, such as x y +=1 2.2.2 x00y yy− 1 yy21− = 2.2.3 xx− 1 xx21− which can also be written x y 1 x1 y1 1 = 0 2.2.4 x2 y2 1 x cos θ + y sin θ = p 2.2.5 2 The four forms are illustrated in figure II.1. y y slope = m y 0 c x x x0 2.2.1 2.2.2 y y • (x2 , y2 ) • ( x1 , y1 ) p x α x 2.2.3 or 4 2.2.5 FIGURE II.1 A straight line can also be written in the form Ax + By + C = 0.
    [Show full text]
  • Geometrical Constructions 1) (See Geometrical Constructions 1)
    GEOMETRICAL CONSTRUCTIONS 2 by Johanna PÉK PhD Budapest University of Technology and Economics Faculty of Architecture Department of Architectural Geometry and Informatics Table of contents • Radical line, radical center. Apollonius’ problem • Conic sections • Basic constructions on ellipses • Basic constructions on parabolas. Basic constructions on hyperbolas • Approximate constructions. Constructions of algebraic operations • Metric problems in space • Polyhedra. Regular polyhedra • Cone, cylinder, sphere • Multi-view representation • Axonometrical representation 2 Power of a point Power of a point (or circle power) The power of a point P with respect to a circle is defined by the product PA ∙ PB where AB is a secant passing through P. Remark: The point P can be an inner point of the circle. 2 Theorem: Using the notation of the figure, PA ∙ PB = PK ∙ PL = PT Sketch of the proof 3 RADICAL LINE, RADICAL CENTER – 1 Radical line (of two circles) is the locus of points of equal (circle) power with respect to two nonconcentric circles. This line is perpendicular to the line of centers. If two circles have common points K, L then their radical line is line KL. Alternative definition: radical line is the locus of points at which tangents drawn to both circles have the same length. (See point P.) lecture & practical 4 Radical line, radical center – 2 Radical center (of three circles): The radical lines of three circles intersect each other in a special pont called the radical center (or power center). lecture & practical 5 Radical line, radical center – 3 The radical line exists even if the two circles have no common points.
    [Show full text]
  • An Introduction to Convex and Discrete Geometry Lecture Notes
    An introduction to convex and discrete geometry Lecture Notes Tomasz Tkocz∗ These lecture notes were prepared and written for the undergraduate topics course 21-366 An introduction to convex and discrete geometry that I taught at Carnegie Mellon University in Fall 2019. ∗Carnegie Mellon University; [email protected] 1 Contents 1 Introduction 5 1.1 Euclidean space . .5 1.2 Linear and affine hulls . .7 1.3 Exercises . 10 2 Basic convexity 12 2.1 Convex hulls . 12 2.2 Carath´eodory's theorem . 14 2.3 Exercises . 16 3 Separation 17 3.1 Supporting hyperplanes . 17 3.2 Separation theorems . 21 3.3 Application in linear optimisation { Farkas' lemma . 24 3.4 Exercises . 25 4 Further aspects of convexity 26 4.1 Extreme points and Minkowski's theorem . 26 4.2 Extreme points of polytopes . 27 4.3 Exercises . 29 5 Polytopes I 30 5.1 Duality . 31 5.2 Bounded polyhedra are polytopes . 36 5.3 Exercises . 38 6 Polytopes II 39 6.1 Faces . 39 6.2 Cyclic polytopes have many faces . 40 6.3 Exercises . 45 7 Combinatorial convexity 46 7.1 Radon's theorem . 46 7.2 Helly's theorem . 47 7.3 Centrepoint . 49 7.4 Exercises . 51 8 Arrangements and incidences 52 8.1 Arrangements . 52 8.2 Incidences . 54 2 8.3 The crossing number of a graph . 56 8.4 Proof of the Szemer´edi-Trotter theorem . 58 8.5 Application in additive combinatorics . 59 8.6 Exercises . 61 9 Volume 62 9.1 The Brunn-Minkowski inequality . 62 9.2 Isoperimetric and isodiametric inequality .
    [Show full text]
  • Hyperbola Notes for IIT JEE, Download PDF!!!
    www.gradeup.co One of the most important topics from JEE point of view is the conic section. Further ellipse and hyperbola constitute of 2-3 questions every year. Read and revise all the important topics from ellipse and hyperbola. Download the pdf of the Short Notes on Ellipse and Hyperbola from the link given at the end of the article. 1.Basic of conic sections Conic Section is the locus of a point which moves such that the ratio of its distance from a fixed point to its perpendicular distance from a fixed line is always constant. The focus is the fixed point, directrix is the fixed straight line, eccentricity is the constant ratio and axis is the line passing through the focus and perpendicular to the directrix. The point of intersection with axis is called vertex of the conic. 1.1 General Equation of the conic PS2 = e2 PM2 Simplifying above, we will get ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, this is the general equation of the conic. 1.2 Nature of the conic The nature of the conic depends on eccentricity and also on the relative position of the fixed point and the fixed line. Discriminant ‘Δ’ of a second-degree equation is defined as: www.gradeup.co Nature of the conic depends on Case 1: When the fixed point ‘S’ lies on the fixed line i.e., on directrix. Then the discriminant will be 0. Then the general equation of the conic will represent two lines Case 2: When the fixed point ‘S’ does not lies on the fixed line i.e., not on directrix.
    [Show full text]
  • 1 CHAPTER 2 CONIC SECTIONS 2.1 Introduction a Particle Moving
    1 CHAPTER 2 CONIC SECTIONS 2.1 Introduction A particle moving under the influence of an inverse square force moves in an orbit that is a conic section; that is to say an ellipse, a parabola or a hyperbola. We shall prove this from dynamical principles in a later chapter. In this chapter we review the geometry of the conic sections. We start off, however, with a brief review (eight equation-packed pages) of the geometry of the straight line. 2.2 The Straight Line It might be thought that there is rather a limited amount that could be written about the geometry of a straight line. We can manage a few equations here, however, (there are 35 in this section on the Straight Line) and we shall return for more on the subject in Chapter 4. Most readers will be familiar with the equation y = mx + c 2.2.1 for a straight line. The slope (or gradient) of the line, which is the tangent of the angle that it makes with the x-axis, is m, and the intercept on the y-axis is c. There are various other forms that may be of use, such as x y 1 2.2.2 x0 y0 y y y y 1 2 1 2.2.3 x x1 x2 x1 which can also be written x y 1 x1 y1 1 = 0 2.2.4 x2 y2 1 x cos + y sin = p 2.2.5 2 The four forms are illustrated in figure II.1. y y slope = m y0 c x x x0 2.2.1 2.2.2 y y (x2 , y2 ) ( x1 , y1 ) p x x 2.2.3 or 4 2.2.5 FIGURE II.1 A straight line can also be written in the form Ax + By + C = 0.
    [Show full text]
  • Director Circle of a Conic Inscribed in a Triangle Author(S): E. P. Rouse Source: the Mathematical Gazette, No
    Director Circle of a Conic Inscribed in a Triangle Author(s): E. P. Rouse Source: The Mathematical Gazette, No. 2 (Jul., 1894), pp. 12-13 Published by: Mathematical Association Stable URL: http://www.jstor.org/stable/3602373 Accessed: 26-11-2015 04:44 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access to The Mathematical Gazette. http://www.jstor.org This content downloaded from 139.184.14.159 on Thu, 26 Nov 2015 04:44:41 UTC All use subject to JSTOR Terms and Conditions 12 12 1THE MilAT'HEIMATICAL GAZETTE then known, and constructed to indicate tlese correctly above referred to: " Of him it may truly be said that for 17,100 years. It lasted, in fact, but a very few he studied more to serve the public than himself, and years. During the troubles of the civil war it was though he was rich in fame and in the promises of cast away as old rubbish, and in 1646 was found by the great, yet he died poor, to the scandal of an Sir Jonas Moore, and deposited in its dilapidated state ungrateful age." G.
    [Show full text]
  • 24Th Conference on Applications of Computer Algebra
    SPECIAL SESIONS Applications of Computer Algebra - ACA2018 Santiago de Compostela 2018 24th Conference on Applications of Computer Algebra June 18–22, 2018 Santiago de Compostela, Spain S8 Dynamic Geometry and Mathematics Education Wednesday Wed 20th, 10:00 - 10:30, Aula 10 − Thierry Dana-Picard: A new approach to automated study of isoptic curves Wed 20th, 10:30 - 11:00, Aula 10 − Roman Hašek: Exploration of dual curves using dynamic geometry and computer algebra system Wed 20th, 11:30 - 12:00, Aula 10 − Setsuo Takato: Programming in KeTCindy with Combined Use of Cinderella and Maxima Wed 20th, 12:00 - 12:30, Aula 10 − Raúl M. Falcón: Discovering properties of bar linkage mechanisms based on partial Latin squares by means of Dynamic Geometry Systems Wed 20th, 12:30 - 13:00, Aula 10 − Philippe R. Richard: Issues and challenges about instrumental proof Wed 20th, 13:00 - 13:30, Aula 10 − Round table: Dynamic Geometry and Computer Algebra Systems in Mathematics instruction 2 Organizers Tomás Recio Universidad de Cantabria Santander, Spain Philippe R. Richard Université de Montréal Montréal, Canada M. Pilar Vélez Universidad Antonio de Nebrija Madrid, Spain Aim and cope Dynamic geometry environments (DGE) have emerged in the last half-century with an ever-increasing impact in mathematics education. DGE enlarges the field of ge- ometric objects subject to formal reasoning, for instance, simultaneous operations with many geometric objects. Today DGE open the possibility of investigating vi- sually and formulating conjectures, comparing objects, discovering or proving rigor- ously properties over geometric constructions, and Euclidean elementary geometry is required to reason about them. Along these decades various utilities have been added to these environments, such as the manipulation of algebraic equations of geometric objects or the auto- mated proving and discovering, based on computer algebra algorithms, of elemen- tary geometry statements.
    [Show full text]