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If three unequal numbers p, q, r are in HP and their squares are in AP ,then find the ratio of p:q:r.

If three unequal numbers p, q, r are in HP and their squares are in AP ,then find the ratio of p:q:r. 

Grade:12th pass

2 Answers

Susmita
425 Points
5 years ago
p,q,r are in HP.so
q=2pr/(p+r)
Or,2pr=q(p+r)
p2,q2,r2 are in AP.so
p2+r2=2q2
or,(p+r)2-2pr=2q2
or,(p+r)2-(p+r)q=2q2
or,(p+r)2-q(p+r)-2q2=0
Solving
Either p+r=2q or p+r=-q.
Take p+r=2q first.So 2pr=2q2.
Now (p-r)2=(p+r)2-4pr=4q2-4q2=0
Or,p=r.This cannot happen because p?q,r are distinct.
So p+r=-q.Then 2pr=-q2
(p-r)2=q2+2q2=3q2
Or,p-r=\pm \sqrt{3} q
Solving this equation with helo of 
p+r=-q you will get
Or,r=\frac{(1\pm \sqrt{3}) q}{2}
And 
Susmita
425 Points
5 years ago
p=\frac{(-1\pm \sqrt{3})}{2}
So
p:q:r=(-1\pm \sqrt{3}):2:(1\pm \sqrt{3})
This is the answer.Hope it helps.If it does please approve..........................................................................................................................................
 

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