If (2, 3, 9), (5, 2, 1), (1, λ, 8) and (λ, 2, 3) are coplanar, then...

Given,

Points $A\left(2, 3, 9\right), B\left(5, 2, 1\right), C\left(1, \lambda, 8\right) \text{\text{ and } } D\left(\lambda, 2, 3\right)$ are coplanar,

Now we know that condition of coplanarity is \left[$\begin{cases}\overset{\to}{AB} & \overset{\to}{AC} & \overset{\to}{AD}\end{cases}$\right] = 0$</p><p>$⇒ \left|$\begin{cases}3 & - 1 & - 8 \\ - 1 & \lambda - 3 & - 1 \\ \lambda - 2 & - 1 & - 6\end{cases}$\right| = 0$</p><p>$⇒ 3\left(\left(- 6\right)\left(\lambda - 3\right) - 1\right) + 1\left(6 + \lambda - 2\right) - 8\left(1 - \left(\lambda - 3\right)\left(\lambda - 2\right)\right) = 0$</p><p>$⇒ 3\left(- 6\lambda + 17\right) + 4 + \lambda - 8 + 8\left(\lambda - 3\right)\left(\lambda - 2\right) = 0$</p><p>$⇒ - 18\lambda + 51 + 4 + \lambda - 8 + 8{\lambda}^{2} - 40\lambda + 48 = 0$</p><p>$⇒ 8{\lambda}^{2} - 57\lambda + 95 = 0$</p><p>So product of roots will be$\dfrac{95}{8}