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Question

The shortest distance from the point (1,2,-1) to the surface of the sphere x2+y2+z2=54 is


A

36

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B

26

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C

6

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D

2

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Solution

The correct option is B

26


Explanation for the correct answer:

x2+y2+z2=54 is the given equation of the sphere

Comparing it with x-a2+y-b2+z-c2=r2 which is the standard equation of a sphere with center a,b,c and radius r

a=b=c=0 and r2=54⇒r=36

Hence, the given equation is of a sphere with center O0,0,0 and radius 36 units.

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

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

x2+y2+z2=12+22+-12=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=xo-xp2+yo-yp2+zo-zp2

⇒OP=0-12+0-22+0+12

⇒OP=6

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

d=r-OP

⇒d=36-6

⇒d=26

Hence, the shortest distance between the surface of the sphere is 26

Hence, option B is the correct answer.


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