Question

Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.

i) x² +y² +z² + x + y + z = 1

ii) 2x² +4xy+ xz−x−3y + 5z+ 3 = 0

iii) x² −y² −z² +xy + 4yz + x = 0

Answer 1

A conicoid given by equation ax²+by²+cz²+2fyz+2gxz+2hxy+2ux+2vy+2wz+d=0 has a point (x₀,y₀,z₀) as a center if

$\begin{cases} ax_ 0 ​ +hy_0 ​ +gz_ 0 ​ +u=0 &\ \\ hx_ 0 ​ +by_ 0 ​ +fz_ 0 ​ +v=0 \\ gx_ 0 ​ +fy_0 ​ +cz_ 0+w=0\\ \end{cases}$

1)

$a=b=c=1, u=v=w=1/2, d=-1, f=g=h=0$

$\begin{cases} x_ 0 ​ +1/2=0 \\ y_ 0 ​ +1/2=0 \\ z_ 0 ​ +1/2=0\\ \end{cases}$

This is the central conicoid with a center (-1/2,-1/2,-1/2)

2)

$a=2, h=2, g=1/2, u=-1/2, v=-3/2, w=5/2, d=3, b=c=f=0$

$\begin{cases} 2x_0 ​ +2y_0 ​ +z_ 0 ​ /2−1/2=0 \\ 2x_ 0 ​ −3/2=0 \\ x_ 0/2+ ​ ​ 5/2=0\\ \end{cases}$

This is not a central conicoid.

3)

$a=1, b=c=-1, h=1/2, f=2, u=1/2, d=g=v=w=0$

$\begin{cases} x_ 0 ​ +y_ 0 ​ /2+1/2=0 \\ x_ 0 ​ /2−y_ 0 ​ +2z_ 0 ​ =0\\ 2y_0 ​ ​ -z_0=0\\ \end{cases}$

$\begin{cases} x_0 ​ =−1/2−z_ 0 ​ /4 \\ −1/4−z_ 0 ​ /8−z_ 0 ​ /2+2z_ 0 ​ =0\\ y_0=z_0/2 ​ ​ \\ \end{cases}$

$\begin{cases} x_0=-9/11 \\ z_0=2/11 ​ ​ \\ y_0=1/11 ​ ​ \\ \end{cases}$

This is the central conicoid with a center  (-6/11,1/11,2/11)