Let S be the set of all real values of such that a plane passing through the points (-^2,1…

Question

Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $(-\lambda^2,1,1)$, $(1,-\lambda^2,1)$ and $(1,1,-\lambda^2)$ also passes through the point $(-1,-1,1)$. Then $S$ is equal to:

Accepted Answer

With $A(-\lambda^2,1,1),\ B(1,-\lambda^2,1),\ C(1,1,-\lambda^2),\ D(-1,-1,1)$, note $\overrightarrow{AB}=(1+\lambda^2)(1,-1,0)$ and $\overrightarrow{AC}=(1+\lambda^2)(1,0,-1)$. Coplanarity of $A,B,C,D$ requires
$$(1+\lambda^2)^2\begin{vmatrix}1&-1&0\1&0&-1\\lambda^2-1&-2&0\end{vmatrix}=0.$$
The inner determinant equals $\lambda^2-3$, so (since $1+\lambda^2\neq0$) $\lambda^2=3\Rightarrow\lambda=\pm\sqrt3$.

Solutions · 1