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A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is `1/x^2+1/y^2+1/z^2=1/p^2`.

Text Solution

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`cosalpha=(OD)/(OC)`
`OC=(3P)/cosalpha`
`OA=(3P)/cosbeta,OB=(3P)/cosgamma`
`A(0,(3p)/cosbeta,0),B(0,0,(3p)/cosgamma),C((3p)/cosalpha,0,0)`
`x=P/cosalpha,y0p/cosbeta,z=p/cosgamma`
`cosalpha=p/x,cosbeta=p/y,cosgamma=p/z`
`cos^2alpha+cos^2beta+cos^2gamma=1`
`p^2/x^2+p^2/y^2+p^2/z^2=1`
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