Question

Identify the following surfaces as an elliptical paraboloid, hyperbolic paraboloid, a hyperboloid of one sheet, a hyperboloid of two sheets, a cone, a circular cylinder, an elliptical cylinder, or a parabolic cylinder, and identify the axis of symmetry as the x-axis, the y-axis, or the z-axis.

a)−4x^2+9y^2−z^2=0
b)9y^2−5x^2−8z^2+1=0
c)5x^2+8y^2−z=0
d)6y^2+6z^2=1
e)−8x^2−4y^2+z^2=1

Answer 1

a) $−4x^2+9y^2−z^2 = 0$ A cone with y-axis as the axis of symmetry

as y can be expressed in terms of x and z, 9y² = 4x² + z²

b) $9y^2−5x^2−8z^2+1=0$ Hyperboloid of One Sheet with the y-axis as the axis of symmetry

c) $5x^2+8y^2−z=0$ Elliptic paraboloid with z-axis as the axis of symmetry

(z = 5x² + 8y²)

d) $6y^2+6z^2=1$ Elliptic cylinder along the plane yz parallel to x- axis i.e. the x-axis as the axis of symmetry

e) $−8x^2−4y^2+z^2=1$ Hyperboloid of Two sheets with z-axis as the axis of symmetry