Central Conicoid

  • A conicoid is said to be symmetric with the point P if the conicoid is symmetric to the origin when the equation of the conicoid is transformed with origin as the center.

  • The origin is the center of the conicoid if ax2 + by2 + cz2 + 2fyz + 2gzx + 2gxy + 2ux + 2vy + 2wz + d = 0 has u = v = w = 0.

  • A conicois S given by the equation F(x, y, z) = ax2 + by2 + cz2 + 2fyz + 2gzx + 2gxy + 2ux + 2vy + 2wz + d = 0 has a point P(x0, y0, z0) as the center iff
    ax0 + hy0 + gz0 + u = 0
    hx0 + by0 + fz0 + v = 0
    gx0 + fy0 + cz0 + w = 0

  • A conicoid has a central conicoid if it has a unique center, it is non central conicoid if it does not has a center or has more than one center.

  • Standard forms of central conicoids
    TypeStandard form
    coneax2 + by2 + cz2 = 0
    imaginary ellipsoidx2/a2 + y2/b2 + z2/c2 = -1
    ellipsoidx2/a2 + y2/b2 + z2/c2 = 1
    hyperboloid of one sheetx2/a2 + y2/b2 - z2/c2 = 1
    x2/a2 - y2/b2 + z2/c2 = 1
    -x2/a2 + y2/b2 + z2/c2 = 1
    hyperboloid of two sheetsx2/a2 - y2/b2 - z2/c2 = 1
    -x2/a2 - y2/b2 + z2/c2 = 1
    -x2/a2 + y2/b2 - z2/c2 = 1
  • The condition for a line to be a tangent to the central conicoid ax2 + by2 + cz2 = 1 at (x0, y0, z0) is ax0α + by0β + cz0γ = 0, where α, β and γ are the direction ratios of the line.

  • The equation of the tangent plane to a central conicoid ax2 + by2 + cz2 = 1 at a point (x0, y0, z0) is axx0 + byy0 + czz0 = 1.

  • The condition that the plane ux + vy + wz = p is a tangent to the central conicoid ax2 + by2 + cz2 = 1 is u2/a + v2/b + w2/z = p2

  • A planar section of a central conicoid is a conic section.



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