If a variable plane, at a distance of 3 units from the origin, intersects the coordinate a…

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 $	riangle ABC$ is:

Accepted Answer

Suppose centroid be (h, k, $l$)

$$ \therefore $$   x - intp $=$ 3h, y - intp $=$ 3k, z - intp $=$ 3$l$

Equation $${x \over {3h}} + {y \over {3k}} + {z \over {3l}} = 1$$

$$ \therefore $$   Distance from (0, 0, 0)

$$\left| {{{ - 1} \over {\sqrt {{1 \over {9{h^2}}} + {1 \over {9{k^2}}} + {1 \over {9{l^2}}}} }}} \right| = 3$$

$$ \Rightarrow $$   $${1 \over {{x^2}}} + {1 \over {{y^2}}} + {1 \over {{z^2}}} = 1$$

Solutions · 1