Question

show that
U= x²-y²/x²+y², V=2xy/x²+y²
are
dependend also find relation blw them.
​

Answer 1

Step-by-step explanation:

u=x2−y2∴∂u∂x=2xand∂u∂y=−2y…(i)v=2xy∴∂v∂x=2yand∂v∂y=2x…(ii)∂z∂x=∂z∂u.∂u∂x+∂z∂v.∂v∂x=(2x)∂z∂u+(2y)∂z∂v…(iii)from(i)and(ii)∂z∂y

Answer 2

U and V are dependent on each other and their values are in the ratio 2xy/(x²-y²).

Given:

U = (x²-y²)/(x²+y²)

V = 2xy/(x²+y²)

To Find:

the relation between U and V

Solution:

Cross multiplying the first equation, we will get-

(x²+y²) =  (x²-y²)/U

similarly, cross multiplying the second equation, we will get-

(x²+y²) =  2xy/V

Now, since the left-hand side of both equations are the same, we can equate the right-hand sides as follows-

(x²-y²)/U = 2xy/V

Again, cross multiplying-

V/U = 2xy/(x²-y²)

Thus, we can see that U and V are dependent on each other and their values are in the ratio 2xy/(x²-y²).

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