Question

A point $P$ moves such that three mutually perpendicular lines $PA,PB$ and $PC$ are drawn from it cutting $x,y$ and $z$ axis at $A,B\text{and}C$ respectively. The volume of tetrahedron $OABC$ is $\frac{4}{3}$ cubic units (where $O$ is the origin). If locus of $P$ is $(x^2+y^2+z^2{)}^{\mu }=(\lambda xyz)$ then which of the following is correct $(\lambda ,\mu \in R)$?

• A. $\lambda -\mu =61$
• B. $\lambda +\mu =67$
• C. $\lambda \mu =192$
• D. $(\lambda {)}^{\frac{1}{\mu }}=4$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $\lambda -\mu =61$
B $\lambda +\mu =67$
C $\lambda \mu =192$
D $(\lambda {)}^{\frac{1}{\mu }}=4$

Let $P(u,v,w),A(a,0,0),B(0,b,0),C(0,0,c)$
$\overrightarrow{AP}\cdot \overrightarrow{BP}=\overrightarrow{BP}\cdot \overrightarrow{CP}=\overrightarrow{CP}\cdot \overrightarrow{AP}=0u^2+v^2+w^2=au+bv.....\left(1\right)u^2+v^2+w^2=bv+cw.....\left(2\right)u^2+v^2+w^2=cw+au.....\left(3\right)$
From $\left(1\right),\left(2\right)\&\left(3\right)\Rightarrow au=bv=cw$
$u^2+v^2+w^2=2au{u}^2+v^2+w^2=2bv{u}^2+v^2+w^2=2cw(u^2+v^2+w^2{)}^3=(8abcuvw)$

Volume of tetrahedron $=\frac{1}{6}abc=\frac{4}{3}\Rightarrow abc=8(u^2+v^2+w^2{)}^3=64uvw(x^2+y^2+z^2{)}^3=64xyz\lambda =64,\mu =3$