Find the equations of the planes passing through the line of intersection of the planes $\displaystyle \vec{r} \cdot(\hat{i}+3 \hat{j})=6$ and $\displaystyle \vec{r} \cdot(3 \hat{i}-\hat{j}-4 \hat{k})=0$, which are at a distance of $\displaystyle 1$ unit from the origin.

Let the equations of required planes are: $\displaystyle \vec{r}\cdot\left(\hat{i}+3\hat{j}\right)-6+\lambda\left[\vec{r}\cdot\left(3\hat{i}-\hat{j}-4\hat{k}\right)\right]=0$or $\displaystyle \vec{r}\cdot[(1+3\lambda)\hat{i}+(3-\lambda)\hat{j}-4\lambda\hat{k}]=6$ for some $\displaystyle \lambda$According to given condition\[\frac{6}{\sqrt{(1+3\lambda)^2+(3-\lambda)^2+(-4\lambda)^2}}=1\]\[36 = 1+9\lambda^2+6\lambda+9+\lambda^2-6\lambda+16\lambda^2\]\[\Rightarrow 26\lambda^2 = 26 \Rightarrow \lambda = \pm 1\]Equations of planes are: $\displaystyle \vec{r}\cdot(4\hat{i}+2\hat{j}-4\hat{k})=6$, $\displaystyle \vec{r}\cdot(-2\hat{i}+4\hat{j}+4\hat{k})=6$