Question

If (x + a)2 + (y + b)? = 4 (ax + by), where

x, a, y, b are real, the value of xy - ab is​

Answer 1

Answer:

(x+a)²+(y+b)²−4ax−4by=0

=>x²+2ax+a²+y²+2by+b²−4ax−4by=0 (∵(a+b)²=a²+2ab+b²)

=>(x²+2ax−4ax+a²)+(y²+2by−4by+b²)=0

=>(x²−2ax+a²)+(y²−2by+b²)=0

=>(x−a)²+(y−b)²=0 (∵(a-b)²=a²-2ab+b²)

Now, as we know that the square of something is always non negative, hence, the sum of 2 non negative numbers can be 0, only if each of them is individually 0. (Given that the numbers are real).

=>(x−a)²=0 and (y−b)²=0

=>x=a and y=b

∴xy−ab=ab−ab=0