Class 12

Math

3D Geometry

Three Dimensional Geometry

A variable plane which remains at a constant distance 3p from the origin cut the coordinate axes at A, B and C. The locus of the centroid of triangle ABC is

- x−1+y−1+z−1=p−1
- x−2+y−2+z−2=p−2
- x+y+z=p
- x2+y2+z2=p2

**Correct Answer: ** Option(b)

**Solution: **[b] Let equation of the variable plane be xa+yb+zc=1 This meets the coordinate axes at A (a, 0, 0), B (0, b, 0) and C (0, 0, c). Let P(α,β,γ)be the centroid of the ΔABC. Then α=a+0+03,β=0+b+03,γ=0+0+c3 ∴a=3α,b=3β,c=3γ (2) Plane (1) is at constant distance 3p form the origin, so 3p=∣∣0a+0b+0c−1∣∣(1a)2+(1b)2+(1c)2−−−−−−−−−−−−−−−−√ ⇒1a2+1b2+1c2=19p2 (3) Form (2) and (3), we get 19α2+19β2+19γ2=19p2 ⇒α−2+β−2+γ−2=p−2 Generalizing α,β,γ, locus of centroid P P(α,β,γ)is x−2+y−2+z−2=p−2