Paraboloids

  • .The standard form of a non-central conicoid is ax2 + by2 + 2wz + d = 0.
    If w = 0, the equation represents a cylinder or a pair of straight lines. If w ≠ 0, the surface represented by the equation is called a paraboloid.

  • The standard equation of a paraboloid is ax2 + by2 = 2wz, w ≠ 0
    . There are two types of paraboloids.
    When a and b are of the same signs we get an elliptic paraboloid.
    When a and b are of opposite signs we get a hyperbolic paraboloid.

  • The condition for a line with direction ratios α, β, γ to be a tangent to the central conicoid ax2+ by2 = 2z at (x0, y0, z0) is ax0α +by0β = γ.

  • The equation of the tangent plane to the paraboloid ax2 + by2 = 2z at a point (x0, y0, z0) is axx0 + byy0 = (z+z0).


  • The condition that the plane ux+vy+wz = p is a tangent plane to the paraboloid ax2 + by2 = 2z is
    u2/a + v2/b + 2wp = 0.


  • The planar section of a paraboloid is a conic section.



Comments