Question

If f(x + y, x - y) = x x y, then find the expression for f(x, y).
10 x² + y²
20x² - y²
4
30 X-
2
4 None of these​

Answer 1

Answer :

f(x , y) = (x² - y²)/4

Solution :

• Given : f(x+y , x-y) = x•y
• To find : f(x , y) = ?

We have ,

f(x+y , x-y) = x•y

Let x + y = u -----(1) and x - y = v ------(2)

Now ,

Adding eq-(1) and (2) , we have ;

=> x + y + x - y = u + v

=> 2x = u + v

=> x = (u + v)/2

Now ,

Subtracting eq-(2) from eq-(1) , we have ;

=> (x + y) - (x - y) = u - v

=> x + y - x + y = u - v

=> 2y = u - v

=> y = (u - v)/2

Now ,

Putting x + y = u , x - y = v , x = (u + v)/2 and y = (u - v)/2 in the given function we get ;

=> f(u , v) = [(u + v)/2]•[(u - v)/2]

=> f(u , v) = (½•½)•[(u + v)•(u - v)]

=> f(u , v) = ¼•(u² - v²)

Now ,

Substituting x in place of u and y in place of v , we get ;

=> f(x , y) = ¼•(x² - y²)

=> f(x , y) = (x² - y²)/4

Hence ,

f(x , y) = (x² - y²)/4