Give reasons for your answers, either with a short proof or counterexample.? 1.#2x+3y=7z# represents a line in three dimensional space. 2.#x=y=z-1# does not intersect cone #x^2+y^2+z^2+2(yz+zx+xy)=0#.

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Answer 1 · 2018-01-17T13:49:06

See below.

1)
#2x+3y=7z rArr 2x+3y-7z=0# represents a plane in #RR^3#

2)
#x^2 + y^2 + z^2 + 2 (y z + z x + x y)=0# and the line #x=y=z-1# intersect because substituting

#x = lambda#
#y = lambda#
#z=lambda+1# we have

#(1+3lambda)^2 = 0 rArr lambda = -1/3#