Answer to Question #86622 in Analytic Geometry for RAKESH DEY

Question #86622
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
1
Expert's answer
2019-03-25T14:54:44-0400

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


{ax0+hy0+gz0+u=0hx0+by0+fz0+v=0gx0+fy0+cz0+w=0\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:


{x0+1/2=0y0+1/2=0z0+1/2=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:


{2x0+2y0+z0/21/2=02x03/2=0x0/2+5/2=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:


{x0+y0/2+1/2=0x0/2y0+2z0=02y0z0=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}

{x0=1/2z0/41/4z0/8z0/2+2z0=0y0=z0/2\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}

{x0=6/11z0=2/11y0=1/11\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)



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment