A plane is at a constant distance $p$ from the origin and meets the coordinate axes in $A, B, C$ . Sows that the locus of the centroid of length $A B C$ is
$x^{- 2} + y^{- 2} + z^{- 2} = 9 p^{- 2}$

Open in App

Solution

$\begin{matrix} L e t t h e e q u a t i o n o f p l a n e \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 p l a n e a t c o n s tan t d i s tan c e P \frac{1}{{(1 / a)}^{2} + {(1 / b)}^{2} + {(1 / c)}^{2}} = {(P)}^{2} \frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} = \frac{1}{p^{2}} p l a n e m e e t c o o r d i n a t e a t A (a, 0, 0) B (0, b, 0) C (0, 0, b) l e t x, y, z, c o r d i n a t e o f c e n t r e x_{1} = \frac{a + 0 + 0}{3} = \frac{a}{3} y_{1} = \frac{0 + b + 0}{3} = \frac{b}{3} z_{1} = \frac{c}{3} a = 3 x, b = 3 y, c = 2 z \frac{1}{{(3 x)}^{2}} + \frac{1}{{(3 y)}^{2}} + \frac{1}{{(3 z)}^{2}} = \frac{1}{p^{2}} x^{- 2} + y^{- 2} + z^{- 2} = 9 p^{- 2} \end{matrix}$

Suggest Corrections

Similar questions

Q. A variable line is at constant distance

^{'} p^{'}

from the origin and meets the coordinate axes in

^{'} A^{'}

and

^{'} B^{'}

. Show that locus of the centroid of

△ O A B

x^{- 2} + y^{- 2} = 9 P^{- 2} .

Q. A variable line is at a constant distance p from the origin and meets coordinate axes in A and B. Show that the locus of the centroid of the

Δ

OAB is

x^{- 2} + y^{- 2} = p^{- 2}

A variable plane at a constant distance p from the origin meets the axes at A,B,C Through A,B,C planes are drawn parallel to the coordinate planes. Show that the locus of their point of the intersection is given by $x^{- 2} + y^{- 2} + z^{- 2} = p^{- 2}$