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Example 24 - If p, q, r are in GP and equations px2 + 2qx + r = 0 - AP and GP mix questions

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Example 24 - Chapter 9 Class 11 Sequences and Series - Part 2

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Example 24 - Chapter 9 Class 11 Sequences and Series - Part 3 Example 24 - Chapter 9 Class 11 Sequences and Series - Part 4 Example 24 - Chapter 9 Class 11 Sequences and Series - Part 5 Example 24 - Chapter 9 Class 11 Sequences and Series - Part 6

 


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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

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.