Question

The vector equation of the plane passing through the planes $r.(i+j+k)=6$ and $r.(2i+3j+4k)=-5$ and the point $(1,1,1)$ is

• A. $r.(20i+23j+26k)=69$
• B. $r.(2i+23j+26k)=69$
• C. $r.(2i+2j+3k)=69$
• D. $r.(20i+3j+26k)=69$

Answer 1

The correct option is A $r.(20i+23j+26k)=69$

The vector equation of plane passing through the intersection of planes $\overrightarrow{r}.\overrightarrow{n_1}=d_1$ and $\overrightarrow{r}.\overrightarrow{n_2}$ and also through the point $x_1,y_1,z_1$ is

$\overrightarrow{r}.(\overrightarrow{n_1}+\lambda \overrightarrow{n_2})=d_1+\lambda d_2$

According to question plane passes through

$\overrightarrow{r}.(\hat{i}+\hat{j}+\hat{k})=6$

comparing it with $\overrightarrow{r}.\overrightarrow{n_1}=d_1$

$\overrightarrow{n_1}=\hat{i}+\hat{j}+\hat{k}$

And

$d_1=6$

Now, other plane by it also passes

$\overrightarrow{r}.(2\hat{i}+3\hat{j}+4\hat{k})=-5$

$=\overrightarrow{r}.(-2\hat{i}-3\hat{j}-4\hat{k})=5$

Comparing it with $\overrightarrow{r}.\overrightarrow{n_2}=d_2$

$\overrightarrow{n_2}=-2\hat{i}-3\hat{j}-4\hat{k}$

And

$d_2=5$

Now, equation of the required plane

$\overrightarrow{r}.\left[\right(\hat{i}+\hat{j}+\hat{k})-\lambda (2\hat{i}+3\hat{j}+4\hat{k}\left)\right]=5\lambda +6$ ----- (i)

Now,

Putting $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$

$(x\hat{i}+y\hat{j}+z\hat{k}).\left[\right(\hat{i}+\hat{j}+\hat{k})-\lambda (2\hat{i}+3\hat{j}+4\hat{k}\left)\right]=5\lambda +6$

$(1-2\lambda )x+(1-3\lambda )y+(1-4\lambda )z=5\lambda +6$ ---- (ii)

Since it passes through $(1,1,1)$

Therefore,

$(1-2\lambda )1+(1-3\lambda )1+(1-4\lambda )1=5\lambda +6$

Hence

$\lambda =\frac{-3}{14}$

Put the value of $\lambda$ in (i)

$\overrightarrow{r}.\left[\right(\hat{i}+\hat{j}+\hat{k})-(\frac{-3}{14}\left)\right(2\hat{i}+3\hat{j}+4\hat{k}\left)\right]=5\left(\frac{-3}{14}\right)+6$

$\overrightarrow{r}.\left[\right(1+\frac{6}{14})\hat{i}+(1+\frac{9}{14})\hat{j}+(1+\frac{12}{14}\hat{k}\left)\right]=\frac{69}{14}$

$\overrightarrow{r}.(20\hat{i}+23\hat{j}+26\hat{k})=69$

So, the required equation of plane is $\overrightarrow{r}.(20\hat{i}+23\hat{j}+26\hat{k})=69$