Solve the Following Questions

Question:

Let $(\lambda, 2,1)$ be a point on the plane which passes through the ponit $(4,-2,2)$. If the plane is perpendicular to the line joining the points

$(-2,-21,29)$ and $(-1,-16,23)$, then

$\left(\frac{\lambda}{11}\right)^{2}-\frac{4 \lambda}{11}-4$ is equal to

Solution:

$\mid \begin{aligned}&\mathrm{A}(-2,-21,29) \\&\mathrm{B}(-1,-16,33)\end{aligned}$

$\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{PQ}}=0$

$\Rightarrow(\hat{i}+5 \hat{j}-6 \hat{k}) \cdot((4-\lambda) \hat{i}-4 \hat{j}+\hat{k})=0$

$\Rightarrow 4-\lambda-20-6=0$

$\Rightarrow \lambda=-22$

$\Rightarrow\left(\frac{\lambda}{11}\right)^{2}-\frac{4 \lambda}{11}-4=4+8-4=8$

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