Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is

A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is \frac{1}{x^{2}}+\frac{1}{y^{2}}+ \frac{1}{z^{2}}= \frac{1}{p^{2}}\cdot

 

 

 

 
 
 
 
 

Answers (1)

let the coordinates of A, B, C are (a,0,0),(0,b,0) and (0,0,c)
\therefore the   equation of plane is
\frac{x}{a}+\frac{y}{b}+\frac{z}{c}= 1---(i)



Since the distance of plane is equal to 3p from the origin
Then , 3p= \frac{\left | \frac{0}{a}+\frac{0}{b}+\frac{0}{c}-1 \right |}{\sqrt{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}}}
\Rightarrow \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}= \frac{1}{9p^{2}}---(ii)
Let the centroid of \triangle ABC be (x,y,z)
= \left ( \frac{a+0+0}{3},\frac{0+b+0}{3} ,\frac{0+0+c}{3}\right )
= \left ( \frac{a}{3}, \frac{b}{3} , \frac{c}{3}\right )
\Rightarrow a= 3x, b= 3y,c= 3z
Putting the value of a,b and c in (ii) we get
\frac{1}{9x^{2}}+\frac{1}{9y^{2}}+\frac{1}{9z^{2}}= \frac{1}{9p^{2}}
\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}= \frac{1}{p^{2}}
Hence this is the required locus of the centroid is :

Posted by

Ravindra Pindel

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Class 12 board exams 2025 to have 50% competency-based questions; no change in 10th
anam-khan

CBSE Class 12 board exams 2024 over; expected result date, trends of last 10 years

Latest Question