Question:medium

A $=(-2,2,3)$, $B=(13,-3,13)$ are two points and $P$ is a variable point such that \[ PA:PB=2:3. \] If $P$ lies on the sphere \[ x^2+y^2+z^2+ux+vy+wz-247=0, \] then \[ u+v+w= \]

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For a fixed ratio \[ PA:PB=m:n, \] use \[ n^2PA^2=m^2PB^2. \] After expansion, the locus is always a sphere (Apollonius sphere). Compare coefficients directly with the given equation.

Updated On: Jul 29, 2026

$24$
$25$
$26$
$27$

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The Correct Option is C

Solution and Explanation

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A \(=(-2,2,3)\), \(B=(13,-3,13)\) are two points and \(P\) is a variable point such that \[ PA:PB=2:3. \] If \(P\) lies on the sphere \[ x^2+y^2+z^2+ux+vy+wz-247=0, \] then \[ u+v+w= \]

Show Hint

The Correct Option is C

Solution and Explanation

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