Distance between Two Parallel Planes

Q. If $p_1,p_2,p_3$ denote the distances of the plane 2x - 3y + 4z + 2 = 0 from the planes 2x - 3y + 4z + 6 = 0, 4x - 6y + 8z + 3 = 0 and 2x - 3y + 4z - 6 = 0 respectively, then

1. $p_1+8p_2-p_3=0$
2. $p_3^2=16p_2^2$
3. $8p_2^2=p_1^2$
4. $p_1+2p_2+3p_3=\sqrt{29}$

Q. Distance between parallel planes 2x-2y+z+3=0 and 4x-4y+2z+5=0 is

1. $\frac{2}{3}$
2. $\frac{1}{3}$
3. $\frac{1}{6}$
4. 2

Q. If the plane $2x-y+2z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4x-2y+4z+\lambda =0$ and $2x-y+2z+\mu =0,$ respectively, then the maximum value of $\lambda +\mu$ is equal to :

1. $5$
2. $9$
3. $13$
4. $15$

Q. If $p_1,p_2,p_3$ denote the distances of the plane 2x - 3y + 4z + 2 = 0 from the planes 2x - 3y + 4z + 6 = 0, 4x - 6y + 8z + 3 = 0 and 2x - 3y + 4z - 6 = 0 respectively, then

1. $p_1+8p_2-p_3=0$
2. $p_3^2=16p_2^2$
3. $8p_2^2=p_1^2$
4. $p_1+2p_2+3p_3=\sqrt{29}$

Q. $\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{a}& \text{The distance between the line}\overrightarrow{r}=(2\hat{i}-2\hat{j}+3\hat{k})+\lambda (\hat{i}-\hat{j}+4\hat{k})& \text{p}& \frac{25}{3\sqrt{14}}\\ & \text{and plane}\overrightarrow{r}.(\hat{i}+5\hat{j}+\hat{k})=5& & \\ \text{b}& \text{The distance between parallel planes}\overrightarrow{r}.(2\hat{i}-\hat{j}+3\hat{k})=4\text{and}& \text{q}& \frac{13}{7}\\ & \overrightarrow{r}.(6\hat{i}-3\hat{j}+9\hat{k})+13=0\text{is}& & \\ \text{c}& \text{The distance of a point (2, 5, -3) from the plane}\overrightarrow{r}.(6\hat{i}-3\hat{j}+2\hat{k})=4& \text{r}& \frac{10}{3\sqrt{3}}\\ & \text{is}& & \\ \text{d}& \text{The distance of the point (1, 0, -3) from the plane}x-y-z-9=0& \text{s}& 7\\ & \text{measured parallel to line}\frac{x-2}{2}=\frac{y+2}{3}=\frac{z-6}{-6}& & \end{array}$

Then which of the following is correct ?

1. $a\to r;b\to p;c\to q;d\to s$
2. $a\to s;b\to p;c\to q;d\to r$
3. $a\to r;b\to q;c\to p;d\to s$
4. $a\to s;b\to q;c\to p;d\to r$