Paraboloid

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

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

• The condition that the plane \(l x+m y+n z=p\) may touch the paraboloid \(a x^{2}+b y^{2}=2 c z\) is given by: \(\dfrac{l^{2}}{a}+\dfrac{m^{2}}{b}+\dfrac{2 n p}{c}=0\) and the point of contact is given by \(\left(\dfrac{-lc}{an},\dfrac{-mc}{bn},\dfrac{-p}{n}\right)\).

• The tangent plane to \(a x^{2}+b y^{2}=2 c z\) 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 \(a x^{2}+b y^{2}=2 c z\) is given by: \(2 z+c\left(\dfrac{1}{a}+\dfrac{1}{b}\right)=0\).

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

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

  \(\alpha= \dfrac{f}{1+a r}\), \(\beta=\dfrac{g}{1+b r}\), \(\gamma=h+c r\),

  where \(r\) is obtained by solving the equation \(a \dfrac{f^{2}}{(1+a r)^{2}}+b \dfrac{g^{2}}{(1+b r)^{2}}=2 c(h+c r)\).

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

Paraboloid