Question

If $f(2x + 3y,2x - 3y) = 24xy$ then $f(x,y)$ is
A) $2xy$
B) $2({x^2} - {y^2})$
C) ${x^2} - {y^2}$
D) None of these

Answer 1

Hint: In this question we have two terms inside the function i.e., $2x + 3y$ and $2x - 3y$. To simplify this, we will take dummy variables and proceed to get a function in single variables.

Complete step by step answer:
As per the question we have
 $f(2x + 3y,2x - 3y) = 24xy$ --- (1)
We have assume
$2x + 3y = A$ and,
$2x - 3y = B$ .
So equation (1) can be rewritten as a function of $A$ and $B$
 $f(2x + 3y,2x - 3y) = f(A,B)= 24xy$ ---(2)
Adding (i) and (ii), we get
$2x + 3y + 2x - 3y = A + B$
On adding it gives us
$4x = A + B$ --- (3)
Taking the difference of (i) and (ii), we get
$2x+3y-(2x-3y)=A-B$
$\Rightarrow 2x+3y-2x+3y=A-B$
$\Rightarrow 6y=A-B$ --- (4)
If we observe equation (3), equation (4) and relate it to equation (2),
we can multiply (3) and (4) to get the required relationship between the terms
$4x \times 6y = (A+B) \times (A-B)$
$24xy = A^2 - B^2$
Compare it with equation (2), that is nothing but the function of $f(A,B)=24xy$
$f(A,B) =24xy = A^2-B^2$
$f(A,B) = A^2-B^2$
By putting the values of $x,y$ in place of A,B the above equation, we have:
$f(x,y) = {x^2} - {y^2}$
Therefore, the correct option is option (C) ${x^2} - {y^2}$.

Note:
We should note the algebraic identity ${a^2} - {b^2} = (a + b)(a - b)$
We can cross check our answer :
${(2x + 3y)^2} - {(2x - 3y)^2}$
Now we know the formula that ${(a + b)^2} = {a^2} + {b^2} + 2ab$ and ${(a - b)^2} = {a^2} + {b^2} - 2ab$
By applying the formula we can write:
$4{x^2} + 9{x^2} + 12xy - 4{x^2} - 9{x^2} + 12xy = 24xy$
Hence our answer is correct.