IIT JEE , AIEEE Forum - Mathematics , Vectors - Tetrahedron----3 salutes assured

sid.shah.90

If any point within or without(should be outside but thats what the book says :)) a tetrahedron ABCD is joined to the vertices and AO, BO, CO and DO are produced to cut the planes of the opposite faces in P,Q,R,S respectively then prove that:

OP/AP+ OQ/BQ + OS/DS = 1

elastiboysai

Hi Sidshah

Take O to be the origin, it will simplify matters

leta,b,c,d represent the pos. vectors of pts A, B, C, D

for 4 linearly dependent vectors

we can have non zero scalars such that

la+mb+nc+pd=0

frm dis u get

a= - (mb+nc+pd)/l

now OA and OP are collinear

so OP=kOA=ka

r=k(mb+nc+pd)/l

lr-kmb-knc-kpd=0

now r,b,c,d are non copl.

l-km-kn-kp=o

k=l/m+n+p

elastiboysai

Now use

AP = AO+OP(vector not.)

AP=(k+1)OA

and usin wat we derived b4

OP/AP=k/k+1=l/l+m+n+p

now due 2 the symmetry of the situation,

ull get simlar ratios

OQ/BQ=m/l+m+n+p, OR/CR=n/m+l+n+p

OS/DS=p/l+m+n+p

add all n u get

1

edit: all r vector notations

sid.shah.90

Could u plz follow up with a diagram elasti?

elastiboysai

I am marking only P as oderwise itll look messy

Ask & Discuss Questions with Community & Experts