Question

The equation ${\left|\overrightarrow{r}\right|}^2-\overrightarrow{r}.(2i+4j-2k)-10=0$ represents a

• A. circle
• B. plane
• C. sphere of radius $4$
• D. sphere of radius $3$

Answer 1

The correct option is C sphere of radius $4$

Given $|\overline{r}{|}^2-\overline{r}\cdot (2i+4j-2k)-10=0\left(1\right)$

Let $r=x\hat{i}+y\hat{j}+z\hat{k}|\overline{r}|=\sqrt{x^2+y^2+z^2}{|\overline{r}|}^2=x^2+y^2+z^2$

Substitute $\overline{r}$ and $|\overline{r}{|}^2$ in equation $\left(1\right)$

$x^2+y^2+z^2-(x\hat{i}+y\hat{j}+z\hat{k})\cdot (2i+4j-2k)-10=0x^2+y^2+z^2-2x-4y+2z-10=0x^2+y^2+z^2-2x-4y+2z=10$

Add $6$ on both sides

$x^2+y^2+z^2-2x-4y+2z+6=16{(x-1)}^2+{(y-2)}^2+{(z+1)}^2=4^2$

is in the form ${(x-\alpha )}^2+{(y-\beta )}^2+{(z-\gamma )}^2=r^2$ which is sphere with radius $r$

here, $r=4$, radius of sphere