Prove that the paraboloids: x^2/a_1^2+y^2/b_1^2=(2z)/c_1x2a21+y2b21=2zc1; x^2/a_2^2+y^2/b_2^2=(2z)/c_2x2a22+y2b22=2zc2; x^2/a_3^2+y^2/b_3^2=(2z)/c_3x2a23+y2b23=2zc3 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}