Question

A sphere pf constant radius $k$ passes through origin and meets aes in $A,B,C$. The centroid of the triangle $ABC$ lies on the sphere

• A. $9(x^2+y^2+z^2)=4k^2$
• B. $3(x^2+y^2+z^2)=4k^2$
• C. $x^2+y^2+z^2=4k^2$
• D. none of these

Answer 1

Answer: (A)

$9(x^2+y^2+z^2)=4k^2$

Let the coordinates of the points $A,B,C$ be $(a,0,0),(0,b,0)$ and $(0,0,c)$ respectively.

Then equation of the sphere $OABC$ is

$x^2+y^2+z^2-ax-by-cz=0$

Radius of this sphere is equal to $k$

$\therefore a^2+b^2+c^2=4k^2$ ...(1)

Coordinates $(x,y,z)$ of the centroid of $△ABC$ are given by

$x=\frac{1}{3}a,y=\frac{1}{3}b,z=\frac{1}{3}c\Rightarrow a=3x,b=3y,c=3z$ ...(2)

Eliminating $a,b,c$ from (1) and (2), we have

$9x^2+9y^2+9z^2=4k^2\Rightarrow 9(x^2+y^2+z^2)=4k^2$