A variable plane x + my + nz = p (where , m, n are direction cosines) intersects with co-ordinate axes at points A, B and C respectively show that the foot of normal on the plane from origin is the orthocentre of triangle ABC and hence find the coordinates of circumcentre of triangle ABC.
Text Solution
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Sol. 
+ 
+ 
= 1.
Foot H of normal on plane has co-ordinates ( p, mp, np)
Direction ratios of AH are < p – p/ , mp , np > and
direction ratios of BC are 
Since 
. 0 + (mp) 
+ (np) 
= 0,
hence AH perpendicular to BC similarly CH perpendicular to AB.
Hence, H is the orthocentre.
Moreover, in any triangle G(centroid) divides OH in ratio 1 : 2.
coordinates of G are 
H ≡ ( p, mp, np) ⇒ O ≡ 
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