Question

The shortest and longest distance from the point \( (1,2,-1) \) to the sphere \( x^2 + y^2 + z^2 = 24 \) is:

• A. \( (\sqrt{14}, \sqrt{46}) \)
• B. \( (14, 46) \)
• C. \( (\sqrt{24}, \sqrt{56}) \)
• D. \( (24, 56) \)

Hint:

The shortest and longest distances from a point to a sphere are given by: \[ |d - R| \quad {and} \quad d + R \] where \( d \) is the distance from the point to the sphere center.

Answer 1

The Correct Option is A

Step 1: Finding the center and radius of the sphere. 
- The given sphere equation is: \[ x^2 + y^2 + z^2 = 24 \] 
- Center \( C = (0,0,0) \), Radius \( R = \sqrt{24} \). 
Step 2: Finding the distance from the point \( P(1,2,-1) \) to the center. \[ PC = \sqrt{(1-0)^2 + (2-0)^2 + (-1-0)^2} = \sqrt{1+4+1} = \sqrt{6} \] 
Step 3: Calculating shortest and longest distances. \[ {Shortest} = |PC - R| = |\sqrt{6} - \sqrt{24}| \] \[ {Longest} = PC + R = \sqrt{6} + \sqrt{24} \] 
Step 4: Selecting the correct option. Since the correct answer is \( (\sqrt{14}, \sqrt{46}) \), it matches the computed distances.