Let S be the set of (,) for which i- j+ k, i+2 j+ k, 3 i-4 j+5 k (with -=5) are coplanar.

Question

Let $S$ be the set of $(\lambda,\mu)$ for which $\lambda\hat i-\hat j+\hat k$, $\hat i+2\hat j+\mu\hat k$, $3\hat i-4\hat j+5\hat k$ (with $\lambda-\mu=5$) are coplanar. Then $\sum_{(\lambda,\mu)\in S}80(\lambda^2+\mu^2)$ is:

Accepted Answer

$$\begin{vmatrix}\lambda&-1&1\1&2&\mu\3&-4&5\end{vmatrix}=0,\qquad \lambda-\mu=5$$

Substituting $\lambda=\mu+5$ into the determinant and simplifying gives a quadratic in $\mu$, with roots $\mu=-15$ (so $\lambda=5/4$) and $\mu=-3$ (so $\lambda=2$).

$$\sum 80(\lambda^2+\mu^2) = 80\left(\frac{25}{16}+225\right)+80(4+9) = 1250+1040=2290$$

Solutions · 1