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Let ABC is be a fixed triangle and P be ...

Let ABC is be a fixed triangle and P be veriable point in the plane of triangle ABC. Suppose a,b,c are lengths of sides BC,CA,AB opposite to angles A,B,C, respectively. If `a(PA)^(2) +b(PB)^(2)+c(PC)^(2)` is minimum, then point P with respect to `DeltaABC` is

A

centroid

B

circumcentre

C

orthocenter

D

incentre

Text Solution

Verified by Experts

The correct Answer is:
D
Let `A(x_(1),y_(1)), B(x_(2),y_(2)),C(x_(3),y_(3))` and `P(h,k)` be the pointa. Now, `aAP^(2)+b BP^(2)+c CP^(2)`
`=a[(h-x_(1))^(2)+(k-y_(1))^(2)] +b[(h-x_(2))^(2)+(k-y_(2))^(2)] +c [(h-x_(2))^(2)+(k-y_(3))^(2)]`
`= [h^(2)(a+b+c) -2h(ax_(1)+bx_(2)+cx_(3))+(ax_(1)^(2)+bx_(2)^(2)+cx_(3)^(2))]`
`+[k^(2)(a+b+c)-2k(ay_(1)+by_(2)+cy_(3))+(ay_(1)^(2)+by_(2)^(2)+cy_(3)^(2))]`
which is minimum when ` = (ax_(1)+bx_(2)+cx_(3))/(a+b+c), k =(ay_(1)+by_(2)+cy_(3))/(a+b+c)` So, P is incentre of `DeltaABC`.
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