Answer to Question #86622 in Analytic Geometry for RAKESH DEY

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.
1) x^2+y^2+z^2+x+y+z=1
2) 2x^2+4xy+xz-x-3y+5z+3=0
3) x^2-y^2-z^2+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

$$\begin{cases} ax_0+hy_0+gz_0+u=0 \\ hx_0+by_0+fz_0+v=0 \\ gx_0+fy_0+cz_0+w=0 \end{cases}$$

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

$$\begin{cases} x_0+1/2=0 \\ y_0+1/2=0 \\ z_0+1/2=0 \end{cases}$$

x²+y²+z²+x+y+z=1 - the central conicoid have a center (-1/2,-1/2,-1/2)

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

$$\begin{cases} 2x_0+2y_0+z_0/2-1/2=0 \\ 2x_0-3/2=0 \\ x_0/2+5/2=0 \end{cases}$$

2x²+4xy+xz-x-3y+5z+3=0 - the conicoid not central

For 3) a=1, b=c=-1, h=1/2, f=2, u=1/2, d=g=v=w=0:

$$\begin{cases} x_0+y_0/2+1/2=0 \\ x_0/2-y_0+2z_0=0 \\ 2y_0-z_0=0 \end{cases}$$

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

$$\begin{cases} x_0=-6/11 \\ z_0=2/11 \\ y_0=1/11 \end{cases}$$

x²-y²-z²+xy+4yz+x=0 - the central conicoid have a center (-6/11,1/11,2/11)