Question

If (x/y) = (z/w), then what is (xy + zw)^2 equal to ?

Answer 1

(x/y) = (z/w)

xw = yz

xw - yz = 0

(xw - yz)² = 0

(xw)² + (yz)² = 2xwyz                                .......................................................(1)

Now, (xy + zw)^2 = (xy)² + (wz)² + 2xwyz

(xy + zw)^2 = (xy)² + (wz)² + (xw)² + (yz)² .......................................................(from 1)

                  = (y² + w²) (x² + z²)

Please verify the answer and tell me.

Answer 2

Given:

(x/y) = (z/w)

To Find:

(xy + zw)² equal to

Solution:

We will solve

(x/y) = (z/w)

By cross multiplication

xw = yz

xw-yz = 0

Squarring both sides

(xw-yz)² = 0

(xw)² + (yz)² - 2xwyz = 0    [using identity(a-b)² = a² +b² - 2ab ]

(xw)² + (yz)² = 2xwyz      ................................(i)

Now,

According to the question:

(xy + zw)² = (xy)² + (zw)² +2xyzw   [Using (a+b) = a² +b² +2ab]

                = (xy)² + (zw)² +2xwyz ...........(ii)

Putting the value of  2xwyz in (ii)  

(xy + zw)²  =  (xy)² + (zw)²+ (xw)² + (yz)²

                  = x²(y² + w²) + z²(w²+ y²)     [Taking x² and z² as common]

                  = (x²+z²)(y² +w²)

Hence, the value of  (xy + zw)² is  (x²+z²)(y² +w²).