1. Chapter 9 Class 11 Sequences and Series
2. Serial order wise

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 ๐/๐ = ๐/๐ q2 = pr Solving the equation px2 + 2qx + r = 0 For ax2 + bx + c roots are x = (โ๐ ยฑโ(๐2 โ 4๐๐))/2๐ Here a = p, b = 2q & c = r Hence the roots of equation px2 + 2qx + r = 0 are x = (โ2q ยฑโ(4q2 โ4rp))/2p Putting q2 = pr from (1) x = (โ2q ยฑโ(4q2 โ4rp))/2p = (โ2q ยฑโ(4pr โ4rp))/2p = (โ2๐ยฑ0)/2๐ = (โ2๐)/2๐ = (โq )/p Thus x = (โ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, (โq )/p is a root of dx2 + 2ex + f = 0 Putting x = (โq )/p in dx2 + 2ex + f = 0 d ((โq )/p)^2 + 2e ((โq )/p) + f = 0 d ((โq)^2 )/p2 โ ((2eq )/p) + f = 0 (๐๐^2 )/p2 โ ((2eq )/p) + f = 0 (๐๐2 โ2๐๐๐+๐๐2)/๐2 = 0 dq2 โ 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 โ (d )/p = ๐/๐ โ ๐/๐ i.e. to show (e )/q + ๐/๐ = ๐/๐ + (d )/p To show :- 2๐/๐ = ๐/๐ + ๐/๐ Now, from (2) dq2 โ 2eap + fp2 = 0 Dividing this by pq2 dq2/pq2 โ 2epq/pq2 + fp2/pq2 = 0/pq2 (d )/p โ (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 โด (d )/p, (e )/q, (f )/r are in A.P Hence proved