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conicoid

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conicoid

  1. 1. WELCOME
  2. 2. Chapter 1 conicoid
  3. 3. Conicoid A surface whose equation is of second degree is called a conicoid. i.e., A surface of which every plane section is a conic.
  4. 4. central conicoid A conicoid all of whose chords through the origin are bisected at the origin is called a central conicoid and is represented by the equation of the form ax2+by2+cz2=1
  5. 5. TYPES 1. If all the three signs are positive, then the surface is called ellipsoid; 2. If one of the signs is negative, the surface is called the hyperboloid of one sheet; 3. If two of the signs are negative, the surface is called the hyperboloid of two sheets; 4. If all the three signs are negative, the surface is called the virtual quadric.
  6. 6. Nature of plane sections of central conicoid The plane section of the central conicoid is an ellipse if c2l2m2<(an2+cl2)(bn2+cm2) ; The projection is a parabola if c2l2m2=(an2+cl2)(bn2+cm2) ; The projection is a hyperbola if c2l2m2>(an2+cl2)(bn2+cm2) .
  7. 7. Tangency: A line which meets a conicoid in two coincident points is called the tangent line to the conicoid. The locus of the tangent lines to the conicoid at a point on it is called the tangent plane at that point.
  8. 8. Condition for the plane lx+my+nz = p to be a tangent plane to the conicoidp = ± 𝑙2 𝑎 + 𝑚2 𝑏 + 𝑛2 𝑐 . Condition of a plane lx+my+nz=p to be tangent plane to the ellipsoid p=± 𝑎2 𝑙2 + 𝑏2 𝑚2 + 𝑐2 𝑛2
  9. 9. Enveloping cone The locus of the tangent lines drawn from a given point to a conicoid is a cone and is called a tangent cone or enveloping cone.The given point is called the vertex of the cone.
  10. 10. Enveloping cylinder The locus of tangent line drawn to a conicoid and parallel to a given line is a cylinder and is called enveloping cylinder of the conicoid.
  11. 11. Polar Plane The polar plane of an external point P with respect to a quadric is the plane which contains the points of contact of all the tangent lines drawn from P to the quadric.
  12. 12. Conjugate points and conjugate planes If the polar plane of a point P passes through the point Q , then the polar plane of Q passes through the point P. Two such points are known as conjugate points and two such planes are called conjugate planes .
  13. 13. Polar lines or Conjugate lines Two lines such that the polar plane of any point lying on one line passes through the other line are called conjugate lines or polar lines. Therefore the polar line of any given line is the line of intersection of the polar planes of any two points on the given line.
  14. 14. Paraboloid A solid generated by the rotation of the parabola about its axis of symmetry is called paraboloid. In other words, paraboloid is a solid having more than two non parallel parabolic cross sections.
  15. 15. TYPES (1).Both the terms are of the same sign, the surface of the ellipse is called elliptic paraboloid. (2).The terms are of the opposite signs, the surface is called hyperbolic paraboloid.
  16. 16. Diametralplanes A line through the centre of the conicoid is called a diameter of the conicoid. A plane through the centre of the conicoid is called a diametral plane of the conicoid. The three diametral planes, which are such that each is the diametral plane of the line of intersection of the other two are called conjugate diametral planes.
  17. 17. Submitted by, k.mythili, 1st year B.ed, Dr.Sivanthi aditanar college of education
  18. 18. THANK YOU

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