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ABC is an equilateral triangle where AB ...

ABC is an equilateral triangle where AB = a and P is any point in its plane such that PA = PB + PC. Then `(PA^(2)+PB^(2)+PC^(2))/(a^(2))` is

A

3

B

`(sqrt(3))/(4)`

C

`(3)/(4)`

D

2

Text Solution

Verified by Experts

The correct Answer is:
D
PA = PB + PC and ABC is equiliateral
`therefore` P lies on arc BC of circumcircle of `Delta ABC`

`therefore angle BPC = 120^(@)`
Using cosine rule in `Delta BPC`
`cos 120^(@)=(PB^(2)+PC^(2)-BC^(2))/(2PB.PC)`
`rArr -PB.PC=PB^(2)+PC^(2)-a^(2)`
`rArr 2PB.PC=2a^(2)-2(PB^(2)+PC^(2))`
Again, PA = PB + PC
`rArr PA^(2)=PB^(2)+PC^(2)+2PB.PC`
`rArr PA^(2)=PB^(2)+PC^(2)+2a^(2)-2(PB^(2)+PC^(2))`
`(PA^(2)+PB^(2)+PC^(2))/(a^(2))=2`
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