# Let $\left ( \lambda ,2,1 \right )$ be a point on the plane which passes through the point $\left ( 4 ,-2,2 \right ).$ If the plane is perpendicular to the line joining the points $\left ( -2,-21,29\right )$ and $\left ( -1,-16,23\right ),$ then $\left ( \frac{\lambda }{11} \right )^{2}-\frac{4\lambda }{11}-4$ is equal to _______. Option: 1 4 Option: 2 6 Option: 3 8 Option: 4 10

Let point A be (-2, -21, 29) and point B be (-1, -16, 23)

and let point P be $(\lambda,2,1)$ and point Q be (4, -2, 2)

Point P and Q lies on the plane.

$\\\vec{AB}\cdot \vec{PQ}=0$

$\\\Rightarrow(\hat{\mathrm{i}}+5 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}) \cdot((4-\lambda) \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+\hat{\mathrm{k}})=0 \\ \Rightarrow 4-\lambda-20-6=0 \\ \Rightarrow \lambda=-22$

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

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