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Question

# If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at A, B and C, then the locus of the centroid of ΔABC is

A
1x2+1y2+1z2=3
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B
1x2+1y2+1z2=1
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C
1x2+1y2+1z2=19
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D
1x2+1y2+1z2=9
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Solution

## The correct option is B 1x2+1y2+1z2=1Let the plane equation be ax+by+cz+d=0Distance of the plane from origin is 3, ∴d√a2+b2+c2=3i.e. d2=9(a2+b2+c2) ...(1)Now, the plane would intersect the x axis at A(−da,0,0),y axis at B(0,−db,0) and z axis at C(0,0,−dc)The centroid of this triangle would have the co-ordinates (−d3a,−d3b,−d3c)Let h=−d3a,k=−d3b,l=−d3cUsing equation (1), we can write 19=19h2+19k2+19l2⇒1=1h2+1k2+1l2

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