Through the intersection line of two surfaces x ^ 2 + y ^ 2 + 4Z = 1 and x ^ 2 = y ^ 2 + Z ^ 2, the generatrix is parallel to the z-axis,

Just eliminate Z. that's LIANLI
X & # 178; + Y & # 178; + 4Z = 1 and X & # 178; = y & # 178; + Z & # 178;
From X & # 178; + Y & # 178; + 4Z = 1, we get 4Z = 1-x ^ 2-y ^ 2
So x & # 178; = y & # 178; + Z & # 178; get 16Z ^ 2 = 16x ^ 2 - 16y ^ 2
So ① the square of both sides is brought into ②: get (1-x ^ 2-y ^ 2) ^ 2 = 16x ^ 2 - 16y ^ 2
It can be simplified to get x ^ 4 + y ^ 4 + 1-18x ^ 2 + 14y ^ 2 + 2x ^ 2, y ^ 2 = 0

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