Answer to Question #198835 in Analytic Geometry for Almas Bashir

Question

Explain Quadratic surfaces with diagram and at least one example in each case.

Answer 1

Ans:-

 There are six different quadric surfaces: the ellipsoid, the elliptic paraboloid, the hyperbolic paraboloid, the  cone, and hyperboloids.

Quadric surfaces are the graphs of equations that can be expressed in the form

$Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0$

When a quadric surface intersects a coordinate plane, the trace is a conic section.

There are some example of quadratic surfaces

An ellipsoid is a surface described by an equation of the form $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$

A set of lines parallel to a given line passing through a given curve is called a cylinder, or a cylindrical surface. A cylinder is a surface that consists of all lines (rulings) that are parallel to a given line and pass through a given plane curve. A cylinder is not a quadratic surface.

i.e. $x^2+y^2=1$ has only 2 dimensions.

Cone is a quadratic surface which has expression

$\dfrac{z^2}{c^2}=\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}$