A variable plane cuts the positive x–axis, positive y–axis and positive z–axis at the points A, B and C respectively such that the volume of the tetrahedron OABC remains constant equal to 32 cubic units and O is the origin of the co–ordinate system.
LIST-I LIST-II
A) The equation to the locus of the centroid of the tetrahedron is P) xyz = 24
B) The equation to the locus of the point equidistant form O, A, B and C is Q) ${(x^{2} + y^{2} + z^{2})}^{3} = 192 x y z$
C) The equation to the locus of the foot of perpendicular from origin to the plane is R) xyz = 3
D) If PA, PB and PC are mutually perpendicular then the locus of P is S) ${(x^{2} + y^{2} + z^{2})}^{3} = 1536 x y z$

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

A–Q; B–R; C–P; D–S

A–P; B–Q; C–R; D–S

A–R; B–P; C–Q; D–S

A–Q; B–R; C–S; D–P

answer is C.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

Given $\frac{a b c}{6} = 32$
Where $A = (a, 0, 0), B (0, b, c), C (0, 0, c)$
a) Centroid of tetrahedron $(α, β, δ) = (\frac{a}{4}, \frac{b}{4}, \frac{c}{4})$
$\Rightarrow 64 α β δ = a b c \Rightarrow x y z = 3$
b) Equidistant point $(α, β, δ) = (\frac{a}{2}, \frac{b}{2}, \frac{c}{2})$
$\Rightarrow 8 α β δ = a b c \Rightarrow x y z = 24$
c) The equation of the plane is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$
$∴$ Foot of the perpendicular from origin = $(α, β, δ)$

$(α, β, δ) = (\frac{1 / a}{Σ \frac{1}{a^{2}}}, \frac{1 / b}{Σ \frac{1}{a^{2}}}, \frac{1 / c}{Σ \frac{1}{a^{2}}})$ $\Rightarrow {(α^{2} + β^{2} + δ^{2})}^{3} = 192 α β δ$
D) Let P be $(α, β, δ)$ then $P A ⊥ P B \Rightarrow α (α - a) + β (β - b) + δ δ = 0$
$\Rightarrow a α + b β = α^{2} + β^{2} + δ^{2}$

$P B ⊥ P C \Rightarrow b β + c δ = α^{2} + β^{2} + δ^{2}$

Now $a b c = 32 \times 6 \Rightarrow {(α^{2} + β^{2} + δ^{2})}^{3} = 1536 α β δ$

Watch 3-min video & get full concept clarity

tricks from toppers of Infinity Learn

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET

🔥 Start Your JEE/NEET Prep at Just ₹1999 / month - Limited Offer! Check Now!

Get Expert Academic Guidance – Connect with a Counselor Today!

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET

	LIST-I		LIST-II
A)	The equation to the locus of the centroid of the tetrahedron is	P)	xyz = 24
B)	The equation to the locus of the point equidistant form O, A, B and C is	Q)	${(x^{2} + y^{2} + z^{2})}^{3} = 192 x y z$
C)	The equation to the locus of the foot of perpendicular from origin to the plane is	R)	xyz = 3
D)	If PA, PB and PC are mutually perpendicular then the locus of P is	S)	${(x^{2} + y^{2} + z^{2})}^{3} = 1536 x y z$