Question

If (5xy + 3y)^2- (5xy - 3y)^2 = Kxy^2, = then the value of 'K' is:​

Answer 1

◕ Question :

If $\sf{(5xy + 3y)^{2} - (5xy - 3y)^{2} = kxy^{2}}$, then, find the Value of "k".

◕ To Find :

The value of "k" in the Expression.

◕ We Know :

• $\sf{(a + b)^{2} = a^{2} + 2ab + b^{2}}$

• $\sf{(a - b)^{2} = a^{2} - 2ab + b^{2}}$

◕ Solution :

$\purple{\sf{(5xy + 3y)^{2} - (5xy - 3y)^{2} = kxy^{2}}}$

Using the identity ,

• $\sf{(a + b)^{2} = a^{2} + 2ab + b^{2}}$

• $\sf{(a - b)^{2} = a^{2} - 2ab + b^{2}}$

We Get :

$\sf{\Rightarrow (5xy)^{2} + 2 \times 5xy \times 3y + (3y)^{2} - \big((5xy)^{2} - 2 \times 5xy \times 3y + (3y)^{2}\big) = kxy^{2}}$

$\sf{\Rightarrow 25x^{2}y^{2} + 2 \times 15xy^{2} + 9y^{2} - \big(25x^{2}y^{2} - 2 \times 15xy^{2} + 9y^{2}\big) = kxy^{2}}$

$\sf{\Rightarrow 25x^{2}y^{2} + 30xy^{2} + 9y^{2} - \big(25x^{2}y^{2} - 30xy^{2} + 9y^{2}\big) = kxy^{2}}$

$\sf{\Rightarrow \cancel{25x^{2}y^{2}} + 30xy^{2} + \cancel{9y^{2}} - \cancel{25x^{2}y^{2}} + 30xy^{2} - \cancel{9y^{2}} = kxy^{2}}$

$\sf{\Rightarrow 30xy^{2} + 30xy^{2} = kxy^{2}}$

$\sf{\Rightarrow 60xy^{2} = kxy^{2}}$

$\sf{\Rightarrow \dfrac{60\cancel{xy^{2}}}{\cancel{xy^{2}}} = k}$

$\purple{\sf{\Rightarrow 60 = k}}$

Hence ,the value of "k" in the Equation is 60 .

$\purple{\sf{\underline{\boxed{(5xy + 3y)^{2} - (5xy - 3y)^{2} = 60xy^{2}}}}}$

» Additional information :

• $\sf{a^{2} - b^{2} = (a + b)(a - b)}$

• $\sf{a^{2} + b^{2} = (a + b)^{2} - 2ab}$

• $\sf{a^{2} + \dfrac{1}{x^{2}} = \bigg(a + \dfrac{1}{x}\bigg)^{2} - 2ab}$

• $\sf{a^{2} + \dfrac{1}{x^{2}} = \bigg(a - \dfrac{1}{x}\bigg)^{2} + 2ab}$

Answer 2

Step-by-step explanation:

$\bf{\underline{\underline\red{Given:-}}}$

• $\rm {(5xy + 3y)}^{2} - {(5xy - 3y)}^{2} = kx {y}^{2}$

$\bf{\underline{\underline\blue{To\:Find:-}}}$

• The value of ' k '

$\bf{\underline{\underline\green{Solution:-}}}$

To solve this problem you need to know some basic identities.

• $\rm {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab$

• $\rm {(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab$

Now:-

$\rm {(5xy + 3y)}^{2} - {(5xy - 3y)}^{2} = kx {y}^{2}$

$\implies\rm \big \{ {(5xy) } ^{2} + {(3y)}^{2} + 2(5xy)(3y) \big\} - \big \{{(5xy) } ^{2} + {(3y)}^{2} - 2(5xy)(3y) \big \} = kx {y}^{2}$

$\implies\rm \big \{ 25 {x}^{2} {y}^{2} + {9y}^{2} + 30x {y}^{2} \big\} - \big \{ 25 {x}^{2} {y}^{2} + {9y}^{2} - 30x {y}^{2}\big \} = kx {y}^{2}$

$\implies\rm 25 {x}^{2} {y}^{2} + {9y}^{2} + 30x {y}^{2} - 25 {x}^{2} {y}^{2} - {9y}^{2} + 30x {y}^{2} = kx {y}^{2}$

$\implies\rm 25 {x}^{2} - 25 {x}^{2} {y}^{2} + {9y}^{2} - {9y}^{2} + 30x {y}^{2} + 30x {y}^{2}= kx {y}^{2}$

$\implies\rm 60x {y}^{2}= kx {y}^{2}$

$\implies\rm \dfrac{60x {y}^{2}}{x {y}^{2} }= k$

$\implies\rm 60= k$

$\implies\rm k = 60$

The value of k = 60

$\bf{\underline{\underline\pink{Alternatively:-}}}$

By using the identity

• $\rm {(a + b)}^{2} - {(a - b)}^{2} = 4ab$

Here (5xy + 3y)² - (5xy - 3y)² is the form (a + b)² - (a - b)²

Here:-

• a = 5xy

• b = 3y

$\rm {(5xy + 3y)}^{2} - {(5xy - 3y)}^{2} = kx {y}^{2}$

$\rm \implies4(5xy)(3y)= kx {y}^{2}$

$\rm \implies4(15x {y}^{2} )= kx {y}^{2}$

$\rm \implies60x {y}^{2} = kx {y}^{2}$

tex]

$\implies\rm 60= k$

$\implies\rm k = 60$

$\large\underline{ \boxed{\rm \purple{ \therefore The \: value \: of \: k = 60}}}$