Let f : R − { 0 } → ( − ∞ , 1 ) be a polynomial of degree 2, satisfying f ( x ) f ( x 1 ) = f ( x ) + f ( x 1 ) . If f ( K ) = − 2 K , then the sum of squares of all possible values of K is :

Question

Let f : R − { 0 } → ( − ∞ , 1 ) be a polynomial of degree 2, satisfying f ( x ) f ( x 1 ​ ) = f ( x ) + f ( x 1 ​ ) . If f ( K ) = − 2 K , then the sum of squares of all possible values of K is :

TestHub · Accepted Answer

as f ( x ) is a polynomial of degree two let it be f ( x ) = a x 2 + b x + c ( a  = 0 ) on satisfying given conditions we get C = 1& a = ± 1 hence f ( x ) = 1 ± x 2 also range ∈ ( − ∞ , 1 ] hence f ( x ) = 1 − x 2 now f ( k ) = − 2 k 1 − k 2 = − 2 k → k 2 − 2 k − 1 = 0 let roots of this equation be α & β then α 2 + β 2 = ( α + β ) 2 − 2 α β = 4 − 2 ( − 1 ) = 6

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