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In a tetrahedron OABC, the edges are of ...

In a tetrahedron OABC, the edges are of lengths, `|OA|=|BC|=a,|OB|=|AC|=b,|OC|=|AB|=c.` Let `G_1 and G_2` be the centroids of the triangle ABC and AOC such that `OG_1 _|_ BG_2,` then the value of `(a^2+c^2)/b^2` is

A

2

B

3

C

6

D

9

Text Solution

Verified by Experts

The correct Answer is:
B
`vec(OG_(1)).vec(BG_(2))=0`
`rArr (a+vecb+vecc)/(3) . (a+vecc-3vecb)/(3)=0`
Now, `|vecc-veca|^(2)=b^(2), |vecc-vecb|=a^(2)` and `|veca-vecb|=c^(2)`.
`therefore 2veca.vecc=a^(2)+c^(2)-b^(2), 2vecb.vecc=b^(2)+c^(2)=a^(2)`,
`2veca.vecb=a^(2)+b^(2)-c^(2)`
Putting in the above result, we get `2a^(2)+2c^(2)-6b^(2)=0`
`rArr (a^(2)+c^(2))/b^(2)=3`
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