Use Coupon: CART20 and get 20% off on all online Study Material

Total Price: Rs.

There are no items in this cart.
Continue Shopping
Bhanu Kiran Grade: Upto college level

If the sum of the roots of the quadratic equation ax^2+ bx+c=0. is equal to the sum of the squares of their reciprocals, then a/c, b/a, c/b are in ....

7 years ago

Answers : (2)

879 Points

Dear bhanu,

1 Given equation is ax2 + bx + c = 0

2 Let α, β be the roots.

3 α + β = -ba and αβ = ca
 [Sum of the roots = -ba and Product of the roots = ca.]

4 α + β = 1 α2+ 1 β2
 [Given condition.]

5 α + β= α2 + β2 α2 β2

6 α + β = (α + β)² - 2αβα² β² 
 [Use α2 + β2 = (α + β)2 - 2α β.]

7 -ba =b²c² -2cac²a²
 [Substitute the values.]

8 -ba =b² - 2acc²
 [Multiply and divide by a2 and simplify.]

9 -ba =b²c²-2acc²

10 2acc²=b²c²+ba

11 2a2c = ab2 + bc2

12 2ab=bc+ca
 [Divide both sides by abc.]

13 Therefore ab, bc, ca are in A.P


Please feel free to ask your queries here. We are all IITians and here to help you in your IIT JEE preparation.

All the best.

Win exciting gifts by answering the questions on Discussion Forum. So help discuss any query on askiitians forum and become an Elite Expert League askiitian.

Now you score 5+15 POINTS by uploading your Pic and Downloading the Askiitians Toolbar  respectively : Click here to download the toolbar..


Askiitians Expert

Sagar Singh

B.Tech, IIT Delhi

7 years ago
vikas askiitian expert
510 Points

let A,B  be the roots of this equation then,

  A+B=-b/a  & AB=c/a          .................1

now it is given that sum of roots is equal to square of resiprocal of roots


         A+B=1/A2 +1/B2

        A+B=A2+B2 /(AB)2 

        A+B   = [(A+B)2 -2AB]/(AB)2

now putting value of A+B and AB from eq 1

      -b/a =  [b2/a2 -2c/a] /c2/a2

      ab2 +bc2 =2a2c                      ........................2

now let a/c,b/a,c/b are in HP then

           a/b = [b/c + c/a]/2

        ab2 + bc2 =2a2c

this result is similar to eq 2 so these terms are in HP.........



7 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies
  • Complete JEE Main/Advanced Course and Test Series
  • OFFERED PRICE: Rs. 15,900
  • View Details

Ask Experts

Have any Question? Ask Experts

Post Question

Answer ‘n’ Earn
Attractive Gift
To Win!!! Click Here for details