Prove that the paraboloids have a common tangent planes

[debrajr]

Prove that the paraboloids:
$$\frac{x^2}{a_1^2}+\frac{y^2}{b_1^2}=\frac{2z}{c_1}$$;

$$\frac{x^2}{a_2^2}+\frac{y^2}{b_2^2}=\frac{2z}{c_2}$$;

$$\frac{x^2}{a_3^2}+\frac{y^2}{b_3^2}=\frac{2z}{c_3}$$

Have a common tangent plane if:
$$\begin{bmatrix}a_1^2 & a_2^2 & a_3^2\\ b_1^2 & b_2^2 & b_3^2\\ c_1 & c_2 & c_3\end{bmatrix}=0$$
Here
$$a_i, b_i, c_i \in \Bbb{R} \left\{0\right\}$$

 

[Greg]

Hello debrajr and welcome to MHB! :D

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