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If $$x^n-py^n+qz^n$$ is divisible by $$x^2+abyz-bzx-axy$$, then what is $$\frac{p}{a^n}-\frac{q}{b^n}$$ equal to?

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Reducing $$x^2+abyz-bzx-axy$$ to the product of two linear factors:





we get x=ay and x=bz as a factor of $$x^2+abyz-bzx-axy$$

Now, since $$x^2+abyz-bzx-axy$$ is a factor of $$x^n-py^n+qz^n$$ 

∴$$ x^n-py^n+qz^n$$ =(x-ay)(x-bz)Q +0, where Q is quotient.

Now, taking LHS,


Put x=ay

=> $$a^ny^n-py^n+qz^n=0 $$  —i)

Put x=bz

=> $$b^nz^n-py^n+qz^n=0 $$  —ii)

eq (ii)-eq (i)

We get, $$a^ny^n=b^nz^n$$


From eq (ii),$$b^nz^n-py^n+qz^n$$=0

Divide by $$b^nz^n, 1-\frac{py^n}{b^nz^n}+\frac{q}{b^n}=0$$ –iii)

Or, $$1-\frac{p}{a^n}+\frac{qz^n}{a^ny^n}=0$$ —iv)     (‘.’ $$a^ny^n=b^nz^n$$)

From (iii) and (iv) we get $$\frac{p}{a^n}-\frac{q}{b^n}$$=1  , Ans 

Answer – (C)