Question

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

• A. $\frac{39}{5}$
• B. $-\frac{39}{5}$
• C. $\frac{44}{5}$
• D. $-\frac{44}{5}$

Answer 1

Answer: (C)

$A (1,5,35), B (7,5,5), C (1, \lambda, 7), D (2 \lambda, 1,2)$
$\overline{ AB }=6 \hat{ i }-30 \hat{ k }, \overline{ BC }=-6 \hat{ i }(\lambda-5) \hat{ j }+2 \hat{ k }$
$\overrightarrow{ CD }=(2 \lambda-1) \hat{ i }+(1-\lambda) \hat{ j }-5 \hat{ k }$
Points are coplanar
$\Rightarrow 0= \begin{vmatrix}6&0&-30\\ -6&\lambda-5&2\\ 2\lambda-1&1-\lambda&-5\end{vmatrix}$
$=6(-5 \lambda+25-2+2 \lambda)$
$-30\left(-6+6 \lambda-\left(2 \lambda^{2}-\lambda-10 \lambda+5\right)\right)$
$=6(-3 \lambda+23)-30\left(-2 \lambda^{2}+11 \lambda-5-6+6 \lambda\right)$
$=6(-3 \lambda+23)-30\left(-2 \lambda^{2}+17 \lambda-11\right)$
$=6\left(-3 \lambda+23+10 \lambda^{2}-85 \lambda+55\right)$
$=6\left(10 \lambda^{2}-88 \lambda+78\right)=12\left(5 \lambda^{2}-44 \lambda+39\right)$
$\Rightarrow 0=12\left(5 \lambda^{2}-44 \lambda+39\right)$
$\lambda_{1}+\lambda_{2}=\frac{44}{5}$