Question:
The equation of the sphere passing through \((1,0,0),(0,1,0),(0,0,1)\) and has its centre on the plane \(x+y+z=6\) is
The equation of the sphere passing through \((1,0,0),(0,1,0),(0,0,1)\) and has its centre on the plane \(x+y+z=6\) is
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For a sphere \(x^2+y^2+z^2+2ux+2vy+2wz+d=0\), centre is \((-u,-v,-w)\).
Updated On: Jul 3, 2026
- \(x^2+y^2+z^2+4x-4y-4z-3=0\)
- \(x^2+y^2+z^2-4x-4y+4z+3=0\)
- \(x^2+y^2+z^2-4x-4y-4z-3=0\)
- \(x^2+y^2+z^2-4x-4y-4z+3=0\)
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The Correct Option is D
Solution and Explanation
Concept:
The general equation of a sphere is \[ x^2+y^2+z^2+2ux+2vy+2wz+d=0 \] Its centre is \[ (-u,-v,-w) \]
Step 1: Use the given points. The sphere passes through \[ (1,0,0) \] Substitute in the general equation: \[ 1+2u+d=0 \] So, \[ 2u+d=-1 \] It passes through \[ (0,1,0) \] So, \[ 1+2v+d=0 \] \[ 2v+d=-1 \] It passes through \[ (0,0,1) \] So, \[ 1+2w+d=0 \] \[ 2w+d=-1 \]
Step 2: Compare the equations. \[ 2u+d=2v+d=2w+d \] Therefore, \[ u=v=w \] Let \[ u=v=w \] Then centre is \[ (-u,-u,-u) \]
Step 3: Use centre condition. The centre lies on \[ x+y+z=6 \] So, \[ (-u)+(-u)+(-u)=6 \] \[ -3u=6 \] \[ u=-2 \] Thus, \[ v=-2,\quad w=-2 \]
Step 4: Find \(d\). Use \[ 1+2u+d=0 \] Substitute \(u=-2\): \[ 1+2(-2)+d=0 \] \[ 1-4+d=0 \] \[ d=3 \]
Step 5: Write the sphere equation. \[ x^2+y^2+z^2+2ux+2vy+2wz+d=0 \] \[ x^2+y^2+z^2+2(-2)x+2(-2)y+2(-2)z+3=0 \] \[ x^2+y^2+z^2-4x-4y-4z+3=0 \]
Step 6: Final answer. \[ \boxed{x^2+y^2+z^2-4x-4y-4z+3=0} \]
The general equation of a sphere is \[ x^2+y^2+z^2+2ux+2vy+2wz+d=0 \] Its centre is \[ (-u,-v,-w) \]
Step 1: Use the given points. The sphere passes through \[ (1,0,0) \] Substitute in the general equation: \[ 1+2u+d=0 \] So, \[ 2u+d=-1 \] It passes through \[ (0,1,0) \] So, \[ 1+2v+d=0 \] \[ 2v+d=-1 \] It passes through \[ (0,0,1) \] So, \[ 1+2w+d=0 \] \[ 2w+d=-1 \]
Step 2: Compare the equations. \[ 2u+d=2v+d=2w+d \] Therefore, \[ u=v=w \] Let \[ u=v=w \] Then centre is \[ (-u,-u,-u) \]
Step 3: Use centre condition. The centre lies on \[ x+y+z=6 \] So, \[ (-u)+(-u)+(-u)=6 \] \[ -3u=6 \] \[ u=-2 \] Thus, \[ v=-2,\quad w=-2 \]
Step 4: Find \(d\). Use \[ 1+2u+d=0 \] Substitute \(u=-2\): \[ 1+2(-2)+d=0 \] \[ 1-4+d=0 \] \[ d=3 \]
Step 5: Write the sphere equation. \[ x^2+y^2+z^2+2ux+2vy+2wz+d=0 \] \[ x^2+y^2+z^2+2(-2)x+2(-2)y+2(-2)z+3=0 \] \[ x^2+y^2+z^2-4x-4y-4z+3=0 \]
Step 6: Final answer. \[ \boxed{x^2+y^2+z^2-4x-4y-4z+3=0} \]
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