Let a,b,c be three distinct numbers, such that a+b+c=xS and a2+b2+c2=S2. If a,b,c are in G.P., then which of the following is/are correct ?
A
13<x2<3
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B
13<x2<1
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C
1<x2<3
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D
13≤x2≤3
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Solution
The correct options are B13<x2<1 C1<x2<3 Let b=ar,c=ar2,r≠±1 as numbers are distinct a2+a2r2+a2r4=S2...(1) a+ar+ar2=xS...(2) Squaring both sides a2(1+r+r2)2=x2S2...(3)
Equation (1) divided by equation (3) ⇒a2(1+r2+r4)a2(1+r+r2)2=S2x2S2 ⇒(1+r2)2−r2(1+r+r2)2=1x2 ⇒1+r2−r21+r+r2=1x2 ⇒x2=r+1r+1r+1r−1
Putting y=r+1r;|y|>2 ⇒x2=y+1y−1 ⇒x2=1+2y−1 Now, when y>2⇒y−1>1⇒0<2y−1<2⇒1<1+2y−1<3⇒1<x2<3