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