The equation of the cone is

x^{2}+2y^{2}-3z^{2}-2yz+5zx+3xy=o (given)

its vertex is ( 0,0,0) and it is homogenous equation x,y,z

the equation of the generator of the cone with direction ( 1,1,-1) are

x-0/1= y-0/1= z-0/-1

Any point on the generator is (r,r,-r)

The equation ate point P will be

x(r)+2y(r)-3z(-r)-[y(-r) +z(r)]+5/2[z(r)+x(-r)]+3/2[x(r)+y(r)]=0

after solving we will get

y+z=0

x^{2}+2y^{2}-3z^{2}-2yz+5zx+3xy=-0

The equation of the cone is

x

^{2}+2y^{2}-3z^{2}-2yz+5zx+3xy=o (given)its vertex is ( 0,0,0) and it is homogenous equation x,y,z

the equation of the generator of the cone with direction ( 1,1,-1) are

x-0/1= y-0/1= z-0/-1

Any point on the generator is (r,r,-r)

The equation ate point P will be

x(r)+2y(r)-3z(-r)-[y(-r) +z(r)]+5/2[z(r)+x(-r)]+3/2[x(r)+y(r)]=0

after solving we will get

y+z=0The equation of the cone is

x

^{2}+2y^{2}-3z^{2}-2yz+5zx+3xy=-0its vertex is ( 0,0,0) and it is homogenous equation x,y,z

the equation of the generator of the cone with direction ( 1,1,-1) are

x-0/1= y-0/1= z-0/-1

Any point on the generator is (r,r,-r)

The equation ate point P will be

x(r)+2y(r)-3z(-r)-[y(-r) +z(r)]+5/2[z(r)+x(-r)]+3/2[x(r)+y(r)]=0

after solving we will get

y+z=0