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Sum of n Terms

If 4 x2-3 y2:...

Question

If (4x²–3y²):(2x²+5y²)=12:19,find x:y

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Solution

We know that ,
If a : b = c : d then a, b , c ,d are in proportion.

a × d = b × d

or

Product of extrems = product of
means

Now ,

( 4x² - 3y²) : ( 2x² + 5y² ) = 12 : 19

( 4x² - 3y² ) 19 = ( 2x² + 5y² )12

76x² - 57y² = 24x² + 60y²

76x² - 24x² = 60y² +57y²

52x² = 117y²

x² / y² = 117 / 52

( x/ y )² = ( 13 × 9 ) / ( 13 × 4 )

= 9 / 4

( x / y )² = ( 3 / 2 )²

Therefore ,

x / y = 3 / 2

x : y = 3 : 2

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(\begin{matrix} 4 x^{2} - 3 y^{2} \end{matrix}) : (\begin{matrix} 2 x^{2} + 5 y^{2} \end{matrix}) = 12 : 19

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({3 x}^{2} - {3 y}^{2}) : ({2 x}^{2} + {5 y}^{2}) = 12 : 19

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Q. Find the product:
(2x² − 5y²) × (x² + 3y²)

(2 x^{2} - 5 y^{2}) \times (x^{2} + 3 y^{2})

x

y

both are positive and

(2 x^{2} - 5 y^{2}) : x y = 1 : 3

x : y

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