Answer
Verified
456k+ 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.
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
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE