Question

If a variable plane, at a distance of \(3\) units from the origin, intersects the coordinate axes at \(A\), \(B\) and \(C\), then the locus of the centroid of \(\Delta ABC\) is :

• A. \(\dfrac1{x^2}+\dfrac1{y^2}+\dfrac1{z^2}=1\)
• B. \(\dfrac1{x^2}+\dfrac1{y^2}+\dfrac1{z^2}=3\)
• C. \(\dfrac1{x^2}+\dfrac1{y^2}+\dfrac1{z^2}=\dfrac19\)
• D. \(\dfrac1{x^2}+\dfrac1{y^2}+\dfrac1{z^2}=9\)

Hint:

Plane: \(\frac xa+\frac yb+\frac zc=1\) with \(A(a,0,0),B(0,b,0),C(0,0,c)\).
Distance from origin \(=\frac{1}{\sqrt{\frac1{a^2}+\frac1{b^2}+\frac1{c^2}}}=3\Rightarrow \frac1{a^2}+\frac1{b^2}+\frac1{c^2}=\frac19\).
Centroid \((x,y,z)=\left(\frac a3,\frac b3,\frac c3\right)\), so \(a=3x,\ b=3y,\ c=3z\).
Then \(\frac1{9x^2}+\frac1{9y^2}+\frac1{9z^2}=\frac19\Rightarrow \frac1{x^2}+\frac1{y^2}+\frac1{z^2}=1\).
Answer: (1) \(\frac1{x^2}+\frac1{y^2}+\frac1{z^2}=1\)

Answer 1

The Correct Option is A

Use the intercept form of the plane; its distance from the origin gives a condition on the intercepts; the centroid is \((a/3,b/3,c/3)\).