Question

Prove that the paraboloids: `x^2/a_1^2+y^2/b_1^2=(2z)/c_1`; `x^2/a_2^2+y^2/b_2^2=(2z)/c_2`; `x^2/a_3^2+y^2/b_3^2=(2z)/c_3` Have a common tangent plane if: `|(a_1^2 a_2^2 a_3^2), (b_1^2 b_2^2 b_3^2), (c_1 c_2 c_3)|=0`?

Prove that the paraboloids:
`x^2/a_1^2+y^2/b_1^2=(2z)/c_1`;

`x^2/a_2^2+y^2/b_2^2=(2z)/c_2`;

`x^2/a_3^2+y^2/b_3^2=(2z)/c_3`

Have a common tangent plane if:
`|(a_1^2 a_2^2 a_3^2), (b_1^2 b_2^2 b_3^2), (c_1 c_2 c_3)|=0`
Here `a_i, b_i, c_i in RR\{0}`

Answer 1

See https://socratic.org/questions/prove-that-the-paraboloids-x-2-a-1-2-y-2-b-1-2-2z-c-1-x-2-a-2-2-y-2-b-2-2-2z-c-2