136. A geometrical interpretation of a system (own).
by Virgil Nicula, Oct 2, 2010, 2:17 PM
Consider 
and the system 
. Prove that the system 
has an unique
solution iff
and in this case its solution is 
.
Consider
with incircle 
and the positive numbers 
, 
, 
which can be the lengths
of the sides of a triangle and for which
. Ascertain the measures
of the angles of if exists the relation 
, where 
.
Proof.
![$m\cdot\underline x^2+[(m+n-p)y-1]\cdot\underline x+ny^2-y+p=0\ (1)$](//latex.artofproblemsolving.com/1/0/0/100ba9baab01915b65b9afc742401504defeb7fe.png)
.
The system admits an unique solution iff ![$\Delta (y)\equiv [(m+n-p)y-1]^2-4m\cdot (ny^2-y+p)=0$](//latex.artofproblemsolving.com/e/5/d/e5d5baa16310d5ea263d6fc804cc96f023f4221a.png)
, i.e.
![$\Delta (y)=\left[\left(m^2+n^2+p^2\right)-2(mn+np+pm)\right]\cdot \underline y^2 +2(m+p-n)\cdot \underline y+(1-4mp)=0$](//latex.artofproblemsolving.com/0/3/e/03ec64e6eeba863d57958cdcb2b7300a72563237.png)
and
![$\Delta\equiv (m+p-n)^2-(1-4mp)\left[\left(m^2+n^2+p^2\right)-2(mn+np+pm)\right]=0$](//latex.artofproblemsolving.com/6/2/b/62b0fc997f524e7d58e624e07af1b96a9f377f8b.png)
, i.e.
and in this case![$\Delta (y)=-[y-(m+p-n)]^2$](//latex.artofproblemsolving.com/d/4/0/d40b36b8a77ccdae2ed290ec09b2b1cfba53f0cc.png)
and 
. Prove easily that and 
, 
.
From the well-known relations 
obtain easily 
i.e.
, 
, 
verify the system with restriction 
. In conclusion, since this system admits an unique solution
obtain that this solution is
, 
, 
a.s.o.
and the system 
. Prove that the system 
has an uniquesolution iff
and in this case its solution is 
.Consider
with incircle 
and the positive numbers 
, 
, 
which can be the lengthsof the sides of a triangle and for which
. Ascertain the measuresof the angles of
if exists the relation 
, where 
. Proof.
![$m\cdot\underline x^2+[(m+n-p)y-1]\cdot\underline x+ny^2-y+p=0\ (1)$](http://latex.artofproblemsolving.com/1/0/0/100ba9baab01915b65b9afc742401504defeb7fe.png)
.The system
admits an unique solution iff ![$\Delta (y)\equiv [(m+n-p)y-1]^2-4m\cdot (ny^2-y+p)=0$](http://latex.artofproblemsolving.com/e/5/d/e5d5baa16310d5ea263d6fc804cc96f023f4221a.png)
, i.e.![$\Delta (y)=\left[\left(m^2+n^2+p^2\right)-2(mn+np+pm)\right]\cdot \underline y^2 +2(m+p-n)\cdot \underline y+(1-4mp)=0$](http://latex.artofproblemsolving.com/0/3/e/03ec64e6eeba863d57958cdcb2b7300a72563237.png)
and![$\Delta\equiv (m+p-n)^2-(1-4mp)\left[\left(m^2+n^2+p^2\right)-2(mn+np+pm)\right]=0$](http://latex.artofproblemsolving.com/6/2/b/62b0fc997f524e7d58e624e07af1b96a9f377f8b.png)
, i.e. and in this case
![$\Delta (y)=-[y-(m+p-n)]^2$](http://latex.artofproblemsolving.com/d/4/0/d40b36b8a77ccdae2ed290ec09b2b1cfba53f0cc.png)
and 
. Prove easily that and 
, 
.
From the well-known relations 
obtain easily 
i.e.
, 
, 
verify the system with restriction 
. In conclusion, since this system admits an unique solutionobtain that this solution is
, 
, 
a.s.o. This post has been edited 17 times. Last edited by Virgil Nicula, Dec 1, 2015, 11:12 AM
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