CGPET 2015 Solved Paper

Let the plane meets the axes at A(a, 0, 0), B(0, b, 0) and C(0, 0, c). Then, equation of plane is

x

a

+

y

b

+

z

c

=1
It is given that plane is at a constant distance p from the origin (0, 0, 0).
∴p=|

0+0−1

√ 1 a2 + 1 b2 + 1 c2

|
⇒

1

p2

=

1

a2

+

1

b2

+

1

c2

..(i)
Let (α, β, γ)be the coordinates of centroid of the formed tetrahedron.
Then, α=

a+0+0+0

4

⇒a=4α
β=

0+b+0+0

4

⇒b=4β
γ=

0+0+c+0

4

⇒c=4γ
On putting values of a, b, c in Eq. (i), we get

1

p2

=

1

(4α)2

+

1

(4β)2

+

1

(4γ)2

⇒

16

p2

=

1

α2

+

1

β2

+

1

γ2

∴ Locus of centroid of tetrahedron is

16

p2

=

1

x2

+

1

y2

+

1

z2