MHB Prove that the paraboloids have a common tangent planes

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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
$$a_i, b_i, c_i \in \Bbb{R} \left\{0\right\}$$
 
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Hello debrajr and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

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