The number of distinct real values of for which the lines x-11=y-22=z+3^2 and x-31=y-2=z-1…

Question

The number of distinct real values of $\lambda$ for which the lines $\dfrac{x-1}1=\dfrac{y-2}2=\dfrac{z+3}{\lambda^2}$ and $\dfrac{x-3}1=\dfrac{y-2}\lambda=\dfrac{z-1}2$ are coplanar is:

Accepted Answer

Coplanarity:
$$\begin{vmatrix}2&0&4\1&2&\lambda^2\1&\lambda&2\end{vmatrix}=0\implies 2(4-\lambda^3)+4(\lambda-2)=0\implies -2\lambda^3+4\lambda=0\implies -2\lambda(\lambda^2-2)=0.$$
So $\lambda=0,\ \sqrt2,\ -\sqrt2$ --- three distinct real values.

Solutions · 1