1.

Solve the equation : `(a)/(x-b)+(b)/(x-a)=2" "(xneb,a)`

Answer» Given equation is
`(a)/(x-b)+(b)/(x-a)=2`
`implies(a)/(x-b)+(b)/(x-a)=1+1`
`implies(a)/(x-b)-1+(b)/(x-a)=1+1`
`implies(a)/(x-b)-1+(b)/(x-a)-1=0`
`implies(a-x+b)/(x-b)+(b-x+a)/(x-a)=0`
`implies(a+b-x)((1)/(x-b)+(1)/(x-a))=0`
`impliesa+b-x=0or(1)/(x-b)+(1)/(x-a)=0`
when `a+b-x=0impliesx=a+b`
and when `(1)/(x-b)+(1)/(x-a)=0implies(x-a+a-b)/((x-b)(x-a))=0`
`implies2x-a-b=0`
`implies2x=a+b`
`impliesx=(a+b)/(2)`
Hence, `x=a+bandx=(a+b)/(2)` are roots of the equaton.
Alternatively,
`implies(1)/(x-b)=-(1)/(a-x)`
`impliesx-b=a-x`
`2x=a+b`
`x=(a+b)/(2)`


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