If (2,3,9), (5,2,1), (1,,8), (,2,3) are coplanar, the product of all possible values of is…

Question

If $(2,3,9)$, $(5,2,1)$, $(1,\lambda,8)$, $(\lambda,2,3)$ are coplanar, the product of all possible values of $\lambda$ is:

Accepted Answer

With $A=(2,3,9)$: $\vec{AB}=(3,-1,-8)$, $\vec{AC}=(-1,\lambda-3,-1)$, $\vec{AD}=(\lambda-2,-1,-6)$.

$$\begin{vmatrix}3&-1&-8\-1&\lambda-3&-1\ \lambda-2&-1&-6\end{vmatrix}=0$$

Expanding gives $8\lambda^2-57\lambda+95=0$, so the product of roots is $\dfrac{95}8$.

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