MCQ
જો બિંદુઓ $(1, 5, 35), (7, 5, 5), (1, \lambda,, 7)$ અને $(2\lambda, 1, 2)$ સમતલીય હોય, તો $\lambda$ ની શક્ય તમામ કિંમતોનો સરવાળો .................. થાય.
- A$\frac{39}{5}$
- B$-\frac{39}{5}$
- C$\frac{44}{5}$
- D$-\frac{44}{5}$
$\overline{ AB }=6 \hat{ i }-30 \hat{ k }, \overline{ BC }=-6 \hat{ i }(\lambda-5) \hat{ j }+2 \hat{ k }$
$\overline{ CD }=(2 \lambda-1) \hat{ i }+(1-\lambda) \hat{ j }-5 \hat{ k }$
Points are coplanar
$\Rightarrow 0=\left|\begin{array}{ccc}6 & 0 & -30 \\ -6 & \lambda-5 & 2 \\ 2 \lambda-1 & 1-\lambda & -5\end{array}\right|$
$=6(-5 \lambda+25-2+2 \lambda)$
$-30\left(-6+6 \lambda-\left(2 \lambda^{2}-\lambda-10 \lambda+5\right)\right)$
$=6(-3 \lambda+23)-30\left(-2 \lambda^{2}+11 \lambda-5-6+6 \lambda\right)$
$=6(-3 \lambda+23)-30\left(-2 \lambda^{2}+17 \lambda-11\right)$
$=6\left(-3 \lambda+23+10 \lambda^{2}-85 \lambda+55\right)$
$=6\left(10 \lambda^{2}-88 \lambda+78\right)=12\left(5 \lambda^{2}-44 \lambda+39\right)$
$\Rightarrow 0=12\left(5 \lambda^{2}-44 \lambda+39\right)$
$\lambda_{1}+\lambda_{2}=\frac{44}{5}$
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