Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

The given planes are $\bar{r}\cdot (2\hat{i}-\lambda \hat{j}+\hat{k})=3$ and $\bar{r}\cdot (4\hat{i}-\hat{j}+\mu \hat{k})=5$.

The normal vectors to the planes are ${\bar{n}}_1=2\hat{i}-\lambda \hat{j}+\hat{k}$ and ${\bar{n}}_2=4\hat{i}-\hat{j}+\mu \hat{k}$.

Since the planes are parallel, their normal vectors are parallel, which means their direction ratios are proportional.

$\frac{2}{4}=\frac{-\lambda }{-1}=\frac{1}{\mu }$

$\frac{1}{2}=\lambda =\frac{1}{\mu }$

From this, we get $\lambda =\frac{1}{2}$ and $\mu =2$.

$\lambda +\mu =\frac{1}{2}+2=\frac{5}{2}$

Answer: $\frac{5}{2}$