Menu Close

Consider-the-system-in-N-3-S-p-2-q-2-r-2-q-p-r-24-r-lt-p-q-Show-that-the-triplet-p-q-r-is-solution-to-S-if-and-only-if-r-lt-12-p-and-q-are-solutions-to-the-equation-n-2-




Question Number 96287 by Ar Brandon last updated on 31/May/20
Consider the system in N^3   (S):  { ((p^2 +q^2 =r^2 )),((q+p+r=24)),((r<p+q)) :}  Show that the triplet (p:q:r) is solution to (S) if  and only if r<12. p and q are solutions to the equation;  n^2 −(24−r)n+24(12−r)=0 where n is an unknown.p
ConsiderthesysteminN3(S):{p2+q2=r2q+p+r=24r<p+qShowthatthetriplet(p:q:r)issolutionto(S)ifandonlyifr<12.pandqaresolutionstotheequation;n2(24r)n+24(12r)=0wherenisanunknown.p
Answered by maths mind last updated on 31/May/20
(p+q)^2 ≥p^2 +q^2 ⇒p+q+r≥2r⇔24≥2r⇒r≤12  (p+q)^2 =(24−r)^2 ⇔p^2 +q^2 +2pq=(24−r)^2   ⇒r^2 +2pq=576−48r+r^2   ⇒pq=288−24r  p+q=24−r⇒(p,q)/solution of  X^2 −(24−r)X+288−24r=0  ⇔n^2 −(24−r)n+12(24−2r)=0
(p+q)2p2+q2p+q+r2r242rr12(p+q)2=(24r)2p2+q2+2pq=(24r)2r2+2pq=57648r+r2pq=28824rp+q=24r(p,q)/solutionofX2(24r)X+28824r=0n2(24r)n+12(242r)=0
Commented by Ar Brandon last updated on 31/May/20
Thank you ��

Leave a Reply

Your email address will not be published. Required fields are marked *