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  • If a,b,c are in AP . and if the equations (b-c)x^2+(c-a)x+(a-b)=0.......(1) and 2(c+a)x^2+(b+c)x=0........(2) have common root , then Show that a^2,c^2,b^2 are in AP

If a,b,c are in AP . and if the equations (b-c)x^2+(c-a)x+(a-b)=0.......(1) and 2(c+a)x^2+(b+c)x=0........(2) have common root , then Show that a^2,c^2,b^2 are in AP

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Solution: Clearly x=1 is a root of(1) .If alpha is the other root of (1),Then

alpha =1 imes alpha =(a-b/b-c)\ \Rightarrow alpha =2a-2b/2b-2c=2a-(a+c)/a+c-2c=1

Thus, the root of (1) are 1,1 Now, (1)and;(2) will have a common root if 1 is also a root of (2)

\ Rightarrow 2(c+a)+b+c=0\ \Rightarrow 2(2b)+b+c=0Rightarrow c=-5b\ \ herefore [a,b,c ;are;in;AP]\ \Rightarrow a=2b-c=2b+5b=7b\ \ herefore a^2=49b^2,c^2=25b^2

Thus a^2,c^2,b^2 are in AP

Posted by

Deependra Verma

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