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Ask community Community Discussion Question: pair of strt lines
Reply Forum Index -> Analytical Geometry originally posted here on IIT-JEE / AIEEE community   
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[Post New] 6 Jul 2007 12:38:07 IST
   Subject: pair of strt lines
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ninad bhat (26)

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if alpha,beta is centroid of triangle whose sides are ax^2+2hxy+by^2=0 and lx+my=1 then prove that alpha/bl-hm=beta/am-hl=2/3*1/bl^2-2hlm+am^2
    
[Post New] 10 Jul 2007 01:41:29 IST
   Subject: Re:pair of strt lines
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gaurav chaudhary (428)

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since ax^2 +2hxy +by^2 = 0  is homogenous eqn second degree we can take two lines as  y=px  ; y=rx
 
lx +my=1
 
pr=a/b
p+r=-2h/b
 
solve to get vertices of triangle
(0,0)
(1/l+mp  , p/l+mp)
(1/l+mr   , r/l+mr)
 
alpha=[2l +m(p+r)]/[3*(l+mp)(l+mr)]
alpha=2*[bl - hm] /[3*(l+mp)(l+mr)]
alpha/bl-hm =2/[3*(l+mp)(l+mr)]
similarly
beta/am-hl =2/[3*(l+mp)(l+mr)]
RHS=2/[3*[l^2+lm(p+r) +m^2.pr]
=2/3*[b.l^2  -2hlm +am^2]
hence proved
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