Just an hour ago, my seniors here were making this paper for a JEE mock test. I, like a good junior, was happily analyzing it for its complexity.
There was a question which i couldn't solve at first sight..(though maybe i could have with pen n paper)
It went like this:
There is a triangle ABC and a point P in space. Specify the point P if PA(square)+PB(square)+PC(square) is to be minimized.
The options were the usual: circumcenter, orthocenter, incenter or centroid??
(And this Digvijay, has a truly 'fantastic', solution)
Would you people care to post your methods too?? in brief?
centroid??
ReplyDeletelet z1, z2, z3 be any 3 points on the argand plane.
cicumcentre as origin for ease
centroid=z1+z2+z3 / 3
orthocentre=(z1+z2+z3)
take any point z4.
mod(z1)=z2=z3(all lie on a circle)
solve and u will get the minimum distance as
3[mod(z1^2) - mod(z4^2)]
you are right with the answer...and probably right with the solution.But... apply some physics mate!!
ReplyDelete(not in the exam..but just for fun?)
I have a "jugadoo" method
ReplyDeleteobviously P can't be circumcentre and orthocentre as they lie outside an obtuse triangle ( ABC is any triangle)
now consider any right angled triangle, suppose with coordinates (0,0),(0,3) and (4,0)
then centroid=(4/3,1) and incentre=(1,1)
on finding PA^2+PB^2+PC^2, it is less for centroid
Ans: centroid
has that physics vala part something to do with gravitation and the fact that COM of a triangle is its centroid?
ReplyDeletenearly there, sambhav..
ReplyDeleteim not getting any nearer...
ReplyDeleteplz tell how's that done?
yaar i'll post the soln. tomorrow.
ReplyDeletebut the point is that u should be able to solve the problem using mathematics..that's important coz thats what really matters..
then u will be able to respect its physical solution.
anyways i have a hint.. the COM of a system of three point masses is at the centroid of the triangel which they form..
moment of inertia is least about COM?
ReplyDeletearre anuj i didn mean it sarcastically yaar...it was just in haste that i wrote it..
i didn't mind man.. just pulling your leg.. anyways you hit it again..
ReplyDeleteenjoyed your TS today? any good problems you would like to share??
also, plz. try to solve this problem using vectors..
thats the approach that matters..
i dnt think 'enjoy' would be the right word(u knw wat i mean!)..nothing unusual yaar..the regular stuff..
ReplyDeletewaise TS mein i was thinking ki how can we photograph a virtual image?(i mean in priciple it isnt correct ryt?)
r phy bhaiya told that COM is the point which has shortest distance to each side relative to other points.
ReplyDeleteso COM of a triangle is centroid n here we get the answer.
???? thats strange. are you sure?
ReplyDeletei mean, what if the mass in the body is non uniformly distributed, lets say increasing with the x coordinate,
isnt the com to be situated somewhere to the right of the geometrical centre???
but i think u might have given me a lead on the question...
centre of mass is point about which moment of inertia is least.
MI is summation of product of mass and square of distance.
assuming mass to be uniformly distributed, summation of squares from the three corner points shuld be minimum about com????
don't assume it to be a triangular plate.. let it be three point masses..
ReplyDeleteOhhhhhhkkkkk.
ReplyDeleteBut question still remains is that how can we assume it to be three point masses?
ohk. just got wat u meant. awesome solution. so u didn need a pen and paper after all!
ReplyDeleteassuming it to be three pt. masses is part of the solution. it gives a physical meaning to the sum of squares..
ReplyDeleteps(for oder ppl)-another useful fact is that KE of a system is least in COM frame(this also came 2 my mind wen i commented above..so thought,wud share)
ReplyDelete