Question

The foot of the perpendicular drawn from the origin to a plane is $(1,-1,5)$ . The equation of the plane is

• A. $\vec{r}.(\hat{i}-\hat{j}+5\hat{k})=27$
• B. $\vec{r}.(\hat{i}-\hat{j}+5\hat{k})=\sqrt{27}$
• C. $\vec{r}.(5\hat{i}-\hat{j}+\hat{k})=\frac{1}{\sqrt{27}}$
• D. $x-y-5z-27=0$

Answer 1

Answer: (A)

$\vec{r}.(\hat{i}-\hat{j}+5\hat{k})=27$

Answer 2

Here, $\vec{n}=\hat{i}-\hat{j}+5\hat{k}$ $\hat{n}=\frac{{\vec{n}}}{|\vec{n}|}=\frac{\hat{i}-\hat{j}+5\hat{k}}{\sqrt{{{(1)}^{2}}+{{(-1)}^{2}}+{{(5)}^{2}}}}$
$\frac{\hat{i}-\hat{j}+5\hat{k}}{\sqrt{27}}$

Also, length of perpendicular from origin to the plane is

$d=\sqrt{{{(1)}^{2}}+{{(-1)}^{2}}+{{(5)}^{2}}}=\sqrt{27}$

Now, equation of plane $\vec{r}.\hat{n}=d$

$\vec{r}.\,\frac{(\hat{i}-\hat{j}+5\hat{k})}{\sqrt{27}}=\sqrt{27}$

$\Rightarrow$ $\vec{r}.\,(\hat{i}-\hat{j}+5\hat{k})=27$