If (1,5,35), (7,5,5), (1,,7), (2,1,2) are coplanar, the sum of all possible values of is:

Question

If $(1,5,35)$, $(7,5,5)$, $(1,\lambda,7)$, $(2\lambda,1,2)$ are coplanar, the sum of all possible values of $\lambda$ is:

Accepted Answer

With $P=(1,5,35)$: $\vec{PQ}=(6,0,-30)$, $\vec{PR}=(0,\lambda-5,-28)$, $\vec{PS}=(2\lambda-1,-4,-33)$.

$$\begin{vmatrix}6&0&-30\0&\lambda-5&-28\2\lambda-1&-4&-33\end{vmatrix}=0$$

Expanding: $6[-33(\lambda-5)-112]+30(2\lambda-1)(\lambda-5)=0 \implies 60\lambda^2-528\lambda+468=0 \implies 5\lambda^2-44\lambda+39=0$

Sum of roots $=\dfrac{44}5$.

Solutions · 1