Answer

Verified

413.4k+ views

**Hint:**For solving this question, we consider the given variables x+2y and x-2y as some other variables X and Y. Then we get two equations in terms of x, y and X, Y. Using those equations, we can find the values of x and y in terms of X and Y. Then we will get the function with variables X and Y only in terms of X and Y, by substituting them in place of x and y. Then we change X as x and Y as y to modify them to get an answer as in the options.

**Complete step by step answer:**

We were given a function f such that $f(x+2y,x-2y)=xy$.

Now let us consider two new variables X, Y such that

X=x+2y

Y=x-2y

Now, let us find the value of x in terms of X and Y.

Let us consider the value of X+Y.

$\begin{align}

& \Rightarrow X+Y=\left( x+2y \right)+\left( x-2y \right) \\

& \Rightarrow X+Y=2x \\

\end{align}$

So, we can write x in terms of X and Y as

$\begin{align}

& \Rightarrow X+Y=2x \\

& \Rightarrow x=\dfrac{X+Y}{2} \\

\end{align}$

Now, let us find the value of y in terms of X and Y.

Let us consider the value of X-Y.

$\begin{align}

& \Rightarrow X-Y=\left( x+2y \right)-\left( x-2y \right) \\

& \Rightarrow X-Y=4y \\

\end{align}$

So, we can write y in terms of X and Y as

$\begin{align}

& \Rightarrow X-Y=4y \\

& \Rightarrow y=\dfrac{X-Y}{4} \\

\end{align}$

So, the values of x and y in terms of X and Y are as below

$x=\dfrac{X+Y}{2}$

$y=\dfrac{X-Y}{4}$

So, we substitute these values in the given function value $f(x+2y,x-2y)=xy$.

Then, we get the function in the terms of X and Y.

$\begin{align}

& \Rightarrow f(x+2y,x-2y)=xy \\

& \Rightarrow f(X,Y)=\left( \dfrac{X+Y}{2} \right)\left( \dfrac{X-Y}{4} \right) \\

& \Rightarrow f(X,Y)=\dfrac{\left( X+Y \right)\left( X-Y \right)}{8} \\

\end{align}$

Now, let us consider the formula,

$\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$

So, the value of product of (X+Y) and (X-Y) is

$\left( X+Y \right)\left( X-Y \right)={{X}^{2}}-{{Y}^{2}}$

Then, we can write the function as,

\[\Rightarrow f(X,Y)=\dfrac{{{X}^{2}}-{{Y}^{2}}}{8}\]

So, for the function f on X and Y value of the function is \[f(X,Y)=\dfrac{{{X}^{2}}-{{Y}^{2}}}{8}\].

Now, let us replace X by x and Y by y, then we can write the function as,

\[\Rightarrow f(x,y)=\dfrac{{{x}^{2}}-{{y}^{2}}}{8}\]

**So, the correct answer is “Option A”.**

**Note:**The chance of occurrence of mistake is at the ending of the solution, one might think that we should not change the variables X and Y into x and y. Here, we are not transforming the variables like we did in the starting of the solution, we are just changing the symbol from X to x and Y to y to make it look like the one in the given options.

Recently Updated Pages

Select the smallest atom A F B Cl C Br D I class 11 chemistry CBSE

Cryolite and fluorspar are mixed with Al2O3 during class 11 chemistry CBSE

The best reagent to convert pent 3 en 2 ol and pent class 11 chemistry CBSE

Reverse process of sublimation is aFusion bCondensation class 11 chemistry CBSE

The best and latest technique for isolation purification class 11 chemistry CBSE

Hydrochloric acid is a Strong acid b Weak acid c Strong class 11 chemistry CBSE

Trending doubts

The provincial president of the constituent assembly class 11 social science CBSE

Gersoppa waterfall is located in AGuyana BUganda C class 9 social science CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The hundru falls is in A Chota Nagpur Plateau B Calcutta class 8 social science CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE