Prove the following

Question:

Let $(\lambda, 2,1)$ be a point on the plane which passes through the point $(4,-2,2) .$ If the plane is perpendicular to the line joining the point $(-2,-21,29)$ and

$(-1,-16,23)$, then

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

Solution:

$\overrightarrow{A B} \perp \overrightarrow{P Q}$

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

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

$N=-22$

Now, $\frac{\lambda}{11}=-2$

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

$\Rightarrow \quad 4+8-4=8$

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