Question:
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
\[
u+v+w=
\]
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
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
- \(24\)
- \(25\)
- \(26\)
- \(27\)
Show Solution
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The Correct Option is C
Solution and Explanation
Concept:
The locus of a point \(P(x,y,z)\) satisfying
\[
\frac{PA}{PB}=\frac{m}{n}
\]
is a sphere (Apollonius sphere) given by
\[
n^2PA^2=m^2PB^2.
\]
Expanding and comparing with the standard sphere equation gives the required coefficients.
Step 1: Use the condition \(PA:PB=2:3\). Given \[ A=(-2,2,3), \qquad B=(13,-3,13). \] Since \[ PA:PB=2:3, \] \[ 3^2PA^2=2^2PB^2. \] \[ 9PA^2=4PB^2. \]
Step 2: Write \(PA^2\) and \(PB^2\). \[ PA^2=(x+2)^2+(y-2)^2+(z-3)^2, \] \[ PB^2=(x-13)^2+(y+3)^2+(z-13)^2. \] Therefore, \[ 9\Big[(x+2)^2+(y-2)^2+(z-3)^2\Big] = 4\Big[(x-13)^2+(y+3)^2+(z-13)^2\Big]. \]
Step 3: Expand both sides. \[ 9(x^2+y^2+z^2+4x-4y-6z+17) \] \[ = 4(x^2+y^2+z^2-26x+6y-26z+347). \] \[ 9x^2+9y^2+9z^2+36x-36y-54z+153 \] \[ = 4x^2+4y^2+4z^2-104x+24y-104z+1388. \]
Step 4: Bring all terms to one side. \[ 5x^2+5y^2+5z^2 +140x -60y +50z -1235 =0. \] Dividing by \(5\), \[ x^2+y^2+z^2 +28x -12y +10z -247 =0. \]
Step 5: Compare with the given sphere. Given sphere: \[ x^2+y^2+z^2+ux+vy+wz-247=0. \] Comparing coefficients, \[ u=28, \qquad v=-12, \qquad w=10. \] Hence, \[ u+v+w = 28-12+10. \] \[ =26. \]
Step 6: Write the final answer. \[ \boxed{26} \]
Step 1: Use the condition \(PA:PB=2:3\). Given \[ A=(-2,2,3), \qquad B=(13,-3,13). \] Since \[ PA:PB=2:3, \] \[ 3^2PA^2=2^2PB^2. \] \[ 9PA^2=4PB^2. \]
Step 2: Write \(PA^2\) and \(PB^2\). \[ PA^2=(x+2)^2+(y-2)^2+(z-3)^2, \] \[ PB^2=(x-13)^2+(y+3)^2+(z-13)^2. \] Therefore, \[ 9\Big[(x+2)^2+(y-2)^2+(z-3)^2\Big] = 4\Big[(x-13)^2+(y+3)^2+(z-13)^2\Big]. \]
Step 3: Expand both sides. \[ 9(x^2+y^2+z^2+4x-4y-6z+17) \] \[ = 4(x^2+y^2+z^2-26x+6y-26z+347). \] \[ 9x^2+9y^2+9z^2+36x-36y-54z+153 \] \[ = 4x^2+4y^2+4z^2-104x+24y-104z+1388. \]
Step 4: Bring all terms to one side. \[ 5x^2+5y^2+5z^2 +140x -60y +50z -1235 =0. \] Dividing by \(5\), \[ x^2+y^2+z^2 +28x -12y +10z -247 =0. \]
Step 5: Compare with the given sphere. Given sphere: \[ x^2+y^2+z^2+ux+vy+wz-247=0. \] Comparing coefficients, \[ u=28, \qquad v=-12, \qquad w=10. \] Hence, \[ u+v+w = 28-12+10. \] \[ =26. \]
Step 6: Write the final answer. \[ \boxed{26} \]
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