Direction ratio of generator is ((x-1),y,(z-1)).
Direction ratio of axis is (1,1,1).
Half angle is 30°.
c o s ( 30 ° ) = ( a 1 a 2 + b 1 b 2 + c 1 c 2 ) / ( a 1 2 + b 1 2 + c 1 2 . a 2 2 + b 2 2 + c 2 2 ) cos (30°)=(a_1a_2+b_1b_2+c_1c_2)/(\sqrt{a_1^{2}+b_1^{2}+c_1^{2}}.\sqrt{a_2^{2}+b_2^{2}+c_2^{2}}) cos ( 30° ) = ( a 1 a 2 + b 1 b 2 + c 1 c 2 ) / ( a 1 2 + b 1 2 + c 1 2 . a 2 2 + b 2 2 + c 2 2 )
3 / 2 = ( x + y + z − 2 ) / ( 3 . ( x − 1 ) 2 + y 2 + ( z − 1 ) 2 ) \sqrt{3}/2=(x+y+z-2)/(\sqrt{3}.\sqrt{(x-1)^{2}+y^{2}+(z-1)^{2}}) 3 /2 = ( x + y + z − 2 ) / ( 3 . ( x − 1 ) 2 + y 2 + ( z − 1 ) 2 )
⟹ 9 [ ( x − 1 ) 2 + y 2 + ( z − 1 ) 2 ] = 4 [ ( x + y + z − 2 ) 2 ] \implies 9[(x-1)^{2}+y^2+(z-1)^{2}]=4[(x+y+z-2)^2] ⟹ 9 [( x − 1 ) 2 + y 2 + ( z − 1 ) 2 ] = 4 [( x + y + z − 2 ) 2 ]
⟹ 5 x 2 + 5 y 2 + 5 z 2 − 8 x y − 8 y z − 8 x z − 2 x − 2 z + 16 y + 2 = 0 \implies 5x^2+5y^2+5z^2-8xy-8yz-8xz-2x-2z+16y+2=0 ⟹ 5 x 2 + 5 y 2 + 5 z 2 − 8 x y − 8 yz − 8 x z − 2 x − 2 z + 16 y + 2 = 0
is the equation of the cone.
Section of the cone by x = 0 x=0 x = 0 is 5 y 2 + 5 z 2 − 8 y z − 2 z + 16 y + 2 = 0 5y^2+5z^2-8yz-2z+16y+2=0 5 y 2 + 5 z 2 − 8 yz − 2 z + 16 y + 2 = 0 .
Section of the cone by y = 0 y=0 y = 0 is 5 x 2 + 5 z 2 − 8 x z − 2 x − 2 z + 2 = 0 5x^2+5z^2-8xz-2x-2z+2=0 5 x 2 + 5 z 2 − 8 x z − 2 x − 2 z + 2 = 0 .
Section of the cone by z = 0 z=0 z = 0 is 5 x 2 + 5 y 2 − 8 x y − 2 x + 16 y + 2 = 0 5x^2+5y^2-8xy-2x+16y+2=0 5 x 2 + 5 y 2 − 8 x y − 2 x + 16 y + 2 = 0 .
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