Paraboloid

• An elliptic paraboloid is represented by the equation ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}={\displaystyle \frac{z}{c}}$, where $c$ is positive.

• A hyperbolic paraboloid is represented by the equation ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}={\displaystyle \frac{z}{c}}$, where $c$ is positive.

• The condition that the plane $lx+my+nz=p$ may touch the paraboloid $a{x}^2+b{y}^2=2cz$ is given by: ${\displaystyle \frac{l^2}{a}}+{\displaystyle \frac{m^2}{b}}+{\displaystyle \frac{2np}{c}}=0$ and the point of contact is given by $({\displaystyle \frac{-lc}{an}},{\displaystyle \frac{-mc}{bn}},{\displaystyle \frac{-p}{n}})$.

• The tangent plane to $a{x}^2+b{y}^2=2cz$ at the point $(\alpha ,\beta ,\gamma )$ is given by $a\alpha x+b\beta y=c(z+\gamma )$.

• The locus of the point of intersection of three mutually perpendicular tangent planes to the paraboloid $a{x}^2+b{y}^2=2cz$ is given by: $2z+c({\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}})=0$.

• The equation of normal at $(\alpha ,\beta ,\gamma )$ is given by ${\displaystyle \frac{x-\alpha }{a\alpha }}={\displaystyle \frac{y-\beta }{b\beta }}={\displaystyle \frac{z-\gamma }{-c}}$.

• The feet of the normals drawn from a point $(f,g,h)$ to the paraboloid $a{x}^2+b{y}^2=2cz$ are given by:

  $\alpha ={\displaystyle \frac{f}{1+ar}}$, $\beta ={\displaystyle \frac{g}{1+br}}$, $\gamma =h+cr$,

  where $r$ is obtained by solving the equation $a{\displaystyle \frac{f^2}{(1+ar{)}^2}}+b{\displaystyle \frac{g^2}{(1+br{)}^2}}=2c(h+cr)$.

  Since this equation is of degree 5, this implies that maximum 5 normals can be drawn from a fixed point to a paraboloid.

Paraboloid