Name: Introduction to 3D Geometry
Uploaded: 2019-07-19
Duration: 10 min 11 s
Description: The world we experience and see around us is a 3D world, and thus, it becomes important to develop our understanding of it. Let us look at the video to under...

Question

A plane meets the coordinate axes in $A, B, C$ such that the centroid of the triangle $A B C$ is the point $(p, q, r)$ . Show that the equation of the plane is $\frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 3$

Accepted Answer

Given centroid of the triangle $A B C$ is $(p, q, r)$

Let the required equation of the plane is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 . . . . (1)$ where $a, b, c$ , are $x -$ intercept, $y -$ intercept, $z -$ intercept respectively.

Thus, coordinates of $A$ , where the plane meets the $x$ -axis are $(a, 0, 0),$ coordinates of $B$ are $(0, b, 0),$ and coordinates of $C$ where the plane cuts the $z$ -axis are $(0, 0, c)$

Centroid of the triangle $A B C$ is $(p, q, r) = (\frac{a + 0 + 0}{3}, \frac{0 + b + 0}{3}, \frac{0 + 0 + c}{3})$

$\Rightarrow (p, q, r) = (\frac{a}{3}, \frac{b}{3}, \frac{c}{3})$ .

We know that two ordered pairs are equal if and only if the corresponding components of the two are equal.

So, $a = 3 p, b = 3 q$ and $c = 3 r$

Now, substitute these values in equation $(1)$

$\Rightarrow \frac{x}{3 p} + \frac{y}{3 q} + \frac{z}{3 r} = 1$

Hence, equation of the plane is $\frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 3$ .

A plane meets the coordinate axes in $A, B, C$ such that the centroid of the triangle $A B C$ is the point $(p, q, r)$ . Show that the equation of the plane is $\frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 3$

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

A plane meets the coordinate axes in A,B,C such that the centroid of the triangle ABC is the point p,q,r. Show that the equation of the plane is xp+yq+zr=3

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

A plane meets the coordinate axes in $A, B, C$ such that the centroid of the triangle $A B C$ is the point $(p, q, r)$ . Show that the equation of the plane is $\frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 3$