1.

If(2a²+3b² ) proportional to ( ab) Then prove that (9a⁴+4b⁴ ) is proportional to ( a²b²).

Answer»

(2a²+3b² )  α  ( ab)

=>(2a²+3b² )  = k  ( ab) [ k is constant]

=>(2a²+3b² )^2  = k^2  ( ab)^2............(1)

=>(2a²+3b² )^2-4*2a^2*3b^2  = k^2  ( ab)^2-24(ab)^2

=>(2a²-3b² )^2  =( k^2 -24)(ab)^2........(2)

Dividing (1) by (2) we get

(2a²+3b² )^2/(2a²-3b² )^2=k^2/(k^2-24) = a constant = m ^2 (say)

So

(2a²+3b² )/(2a²-3b² ) = m 

By componendo and dividendedo we get

(2*2a²)/(2*3b² ) = (m+1)/(m-1)

=>a^2/b^2=3/2*(m+1)/(m-1) =a constant =n  (say)

Now (9a⁴+4b⁴ )/ ( a²b²)

= 9*a^2/b^2+4*b^2/a^2

= 9n+4/n = a constant

So we can say

(9a⁴+4b⁴ ) α  ( a²b²)


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