Last updated at May 29, 2018 by Teachoo

Transcript

Example 24 If p,q,r are in G.P. and the equations, px2 + 2qx + r = 0 and dx2 + 2ex + f = 0 have a common root, then show that (d )/p, (e )/q, (f )/r are in A.P It is given that p, q, r are in G.P So, their common ratio is same / = / q2 = pr Solving the equation px2 + 2qx + r = 0 For ax2 + bx + c roots are x = ( ( 2 4 ))/2 Here a = p, b = 2q & c = r Hence the roots of equation px2 + 2qx + r = 0 are x = ( 2q (4q2 4rp))/2p Putting q2 = pr from (1) x = ( 2q (4q2 4rp))/2p = ( 2q (4pr 4rp))/2p = ( 2 0)/2 = ( 2 )/2 = ( q )/p Thus x = ( q )/p is the root of the equation px2 + 2qx + r = 0 Also, it is given that equations px2 + 2qx + r = 0 & dx2 + 2ex + f = 0 have a common root So, ( q )/p is a root of dx2 + 2ex + f = 0 Putting x = ( q )/p in dx2 + 2ex + f = 0 d (( q )/p)^2 + 2e (( q )/p) + f = 0 d (( q)^2 )/p2 ((2eq )/p) + f = 0 ( ^2 )/p2 ((2eq )/p) + f = 0 ( 2 2 + 2)/ 2 = 0 dq2 2eap + fp2 = 0 We need to show (d )/p, (e )/q, (f )/r are in AP i.e. we need to show their common difference is some i.e. to show :- (e )/q (d )/p = / / i.e. to show (e )/q + / = / + (d )/p To show :- 2 / = / + / Now, from (2) dq2 2eap + fp2 = 0 Dividing this by pq2 dq2/pq2 2epq/pq2 + fp2/pq2 = 0/pq2 (d )/p (2e )/q + (fp )/q2 = 0 (d )/p + (fp )/q2 = (2e )/q Putting q2 = pr from (2) d/p + fp/pr = 2e/q d/p + f/r = 2e/q which is what we have to prove (d )/p, (e )/q, (f )/r are in A.P Hence proved

Examples

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6 Important

Example 7

Example 8

Example 9

Example 10

Example 11

Example 12

Example 13

Example 14 Important

Example 15 Important

Example 16

Example 17 Important

Example 18 Important

Example 19 Important

Example 20

Example 21 Important

Example 22

Example 23 Important

Example 24 Important You are here

Chapter 9 Class 11 Sequences and Series

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.