Question:
The radical plane of the spheres \(x^2+y^2+z^2+4x-2y+2z+6=0\) and \(x^2+y^2+z^2+2x-4y-2z+6=0\) is
The radical plane of the spheres \(x^2+y^2+z^2+4x-2y+2z+6=0\) and \(x^2+y^2+z^2+2x-4y-2z+6=0\) is
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To find the radical plane of two spheres, subtract their equations and simplify.
Updated On: Jul 3, 2026
- \(x-y+2z=0\)
- \(x+y+2z=0\)
- \(x+y-2z=0\)
- \(x-y-2z=0\)
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The Correct Option is B
Solution and Explanation
Concept:
The radical plane of two spheres is obtained by subtracting their equations. If the two spheres are \[ S_1=0 \] and \[ S_2=0 \] then the radical plane is \[ S_1-S_2=0 \]
Step 1: Write the two sphere equations. \[ S_1=x^2+y^2+z^2+4x-2y+2z+6=0 \] \[ S_2=x^2+y^2+z^2+2x-4y-2z+6=0 \]
Step 2: Subtract \(S_2\) from \(S_1\). \[ S_1-S_2=0 \] \[ (x^2+y^2+z^2+4x-2y+2z+6) - (x^2+y^2+z^2+2x-4y-2z+6)=0 \]
Step 3: Cancel common terms. The terms \(x^2,y^2,z^2\) and \(6\) cancel. \[ 4x-2x-2y+4y+2z+2z=0 \] \[ 2x+2y+4z=0 \] Divide by \(2\): \[ x+y+2z=0 \]
Step 4: Final answer. \[ \boxed{x+y+2z=0} \]
The radical plane of two spheres is obtained by subtracting their equations. If the two spheres are \[ S_1=0 \] and \[ S_2=0 \] then the radical plane is \[ S_1-S_2=0 \]
Step 1: Write the two sphere equations. \[ S_1=x^2+y^2+z^2+4x-2y+2z+6=0 \] \[ S_2=x^2+y^2+z^2+2x-4y-2z+6=0 \]
Step 2: Subtract \(S_2\) from \(S_1\). \[ S_1-S_2=0 \] \[ (x^2+y^2+z^2+4x-2y+2z+6) - (x^2+y^2+z^2+2x-4y-2z+6)=0 \]
Step 3: Cancel common terms. The terms \(x^2,y^2,z^2\) and \(6\) cancel. \[ 4x-2x-2y+4y+2z+2z=0 \] \[ 2x+2y+4z=0 \] Divide by \(2\): \[ x+y+2z=0 \]
Step 4: Final answer. \[ \boxed{x+y+2z=0} \]
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