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Question
If the plane 2ax−3ay+4az+6=0 passes through the midpoint of the line joining the centres of the spheres x2+y2+z2+6x−8y−2z=13 and x2+y2+z2−10x+4y−2z=8 then a=
If the plane 2ax−3ay+4az+6=0 passes through the midpoint of the line joining the centres of the spheres x2+y2+z2+6x−8y−2z=13 and x2+y2+z2−10x+4y−2z=8 then a=
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Solution
The correct option is C −2
Given spheres are x2+y2+z2+6x−8y−2z=13 ..... (i) and x2+y2+z2−10x+4y−2z=8 ........ (ii)Writing the above equation of spheres in standard form we get(x+3)2+(y−4)2+(z−1)2=39
and (x−5)2+(y+2)2+(z−1)2=38Centre of spheres are (−3,4,1)and(5,−2,1)
Therefore mid-point(M)=(−3+52,4−22,1+12)=(1,1,1)
Since given plane 2ax−3ay+4az+6=0 passes through the mid-point M
⟹2a(1)−3a(1)+4a(1)+6=0⟹3a=−6
⟹a=−2
Hence, option C is correct.
Given spheres are x2+y2+z2+6x−8y−2z=13 ..... (i)
and x2+y2+z2−10x+4y−2z=8 ........ (ii)
Writing the above equation of spheres in standard form we get
(x+3)2+(y−4)2+(z−1)2=39
and (x−5)2+(y+2)2+(z−1)2=38
Centre of spheres are (−3,4,1)and(5,−2,1)
Therefore mid-point(M)=(−3+52,4−22,1+12)=(1,1,1)
Since given plane 2ax−3ay+4az+6=0 passes through the mid-point M
⟹2a(1)−3a(1)+4a(1)+6=0⟹3a=−6
⟹a=−2
Therefore mid-point(M)=(−3+52,4−22,1+12)=(1,1,1)
Since given plane 2ax−3ay+4az+6=0 passes through the mid-point M
⟹2a(1)−3a(1)+4a(1)+6=0⟹3a=−6
⟹a=−2
Hence, option C is correct.
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