Question

The shortest distance from the point $(1,2,-1)$ to the surface of the sphere $x^2+y^2+z^2=54$ is

• A. $3\sqrt{6}$
• B. $2\sqrt{6}$
• C. $\sqrt{6}$
• D. $2$

Answer 1

The correct option is B

$2\sqrt{6}$

Explanation for the correct answer:

$x^2+y^2+z^2=54$ is the given equation of the sphere

Comparing it with ${\left(x-a\right)}^2+{\left(y-b\right)}^2+{\left(z-c\right)}^2=r^2$ which is the standard equation of a sphere with center $\left(a,b,c\right)$ and radius $r$

$a=b=c=0$ and $r^2=54\Rightarrow r=3\sqrt{6}$

Hence, the given equation is of a sphere with center $O\left(0,0,0\right)$ and radius $3\sqrt{6}$ units.

Consider the point $P(1,2,-1)$

Substitute the co-ordinates of this point in the equation of the given sphere we get

$x^2+y^2+z^2=1^2+2^2+{\left(-1\right)}^2=6<54$

Hence, the point $P(1,2,-1)$ lies inside the sphere

The shortest distance between points $O$ and $P$ is given as

$OP=\sqrt{{\left(x_o-x_p\right)}^2+{\left(y_o-y_p\right)}^2+{\left(z_o-z_p\right)}^2}$

$\Rightarrow$$OP=\sqrt{{\left(0-1\right)}^2+{\left(0-2\right)}^2+{\left(0+1\right)}^2}$

$\Rightarrow$$OP=\sqrt{6}$

Hence, the shortest distance between the surface of the sphere is given as

$d=r-OP$

$\Rightarrow$$d=3\sqrt{6}-\sqrt{6}$

$\Rightarrow$$d=2\sqrt{6}$

Hence, the shortest distance between the surface of the sphere is $2\sqrt{6}$

Hence, option $\left(B\right)$ is the correct answer.