Suppose p,q,r,s are fixed real numbers such that a quadrilateral can be formed with sides p,q,r,s in clockwise order. Prove that the vertices of the quadrilateral of maximum area lie on a circle .
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Area ( Δ ABCD)
= Area of Δ ADB + Area of Δ BDC
A = 
ps sin α + 
qr sin β
= 
ps(+cos α ) + 
qrcos β 
= 0 ⇒ 
= 
BD 2 = p 2 + s 2 – 2pscos α = q 2 + r 2 – 2qr cos β
Differentiating we get
– 2ps (–sin α ) = – 2qr (– sin β ) 
⇒ 
= 
⇒ – 
= 
⇒ sin α cos β + cos α sin β = 0 ⇒ sin( α + β ) = 0 ⇒ α + β = π
Also , 
= 
ps 
= 0 ⇒ α + β = π
If α + β < π then 
> 0
If α + β > π , then 
< 0
∴ By 1st derivative test A has maxima when α + β = π
⇒ A, B, C, D are concyclic
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