Question

The equation of sphere concentric with the sphere \(x^2+y^2+z^2-4x-6y-8z-5=0\) and which passes through the origin, is

• A. \(x^2+y^2+z^2-4x-6y-8z=0\)
• B. \(x^2+y^2+z^2-6y-8z=0\)
• C. \(x^2+y^2+z^2=0\)
• D. \(x^2+y^2+z^2-4x-6y-8z-6=0\)

Hint:

For concentric spheres, keep the linear terms same (same centre). Then use the given point condition (here origin) to find the constant term.

Answer 1

The Correct Option is A

Step 1: Understand what concentric spheres mean.
Concentric spheres have the same centre.
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 2: Find centre of given sphere.
Given:
\[ x^2+y^2+z^2-4x-6y-8z-5=0 \] Compare with \(2u=-4,\ 2v=-6,\ 2w=-8\):
\[ u=-2,\quad v=-3,\quad w=-4 \] So centre is:
\[ (-u,-v,-w)=(2,3,4) \] Step 3: Write equation of concentric sphere.
Since centre must remain \((2,3,4)\), the sphere must have same linear terms:
\[ x^2+y^2+z^2-4x-6y-8z+d=0 \] Step 4: Use condition "passes through origin".
Point \((0,0,0)\) lies on sphere, so substitute:
\[ 0+0+0-0-0-0+d=0 \Rightarrow d=0 \] Step 5: Final equation.
\[ x^2+y^2+z^2-4x-6y-8z=0 \] Final Answer: \[ \boxed{x^2+y^2+z^2-4x-6y-8z=0} \]