QUESTION

# The value of the product of (x-y) and (y-x) is a.${{x}^{2}}+{{y}^{2}}-2xy$b.$2xy+{{x}^{2}}+{{y}^{2}}$c.$2xy-{{x}^{2}}-{{y}^{2}}$d.${{x}^{2}}-2xy+{{y}^{2}}$

Hint: To solve the given question which is calculating the product of (x-y) and (y-x) we proceed using the necessary algebraic calculations and properties of the product of (x-y) with (y-x). To do so we first open one of the brackets, preferably the first one, and then taking the second one as a product of elements of the first bracket. Proceeding in this way we obtain the required result.

We have to calculate the value of the product of (x-y) and (y-x).
Multiplying (x-y) with (y-x) we have,
$(x-y)(y-x)$

Expanding the above by opening from the left and taking x and y simultaneously common we have,
$(x-y)(y-x)=x(y-x)-y(y-x)$

Again, opening the right-hand side of the equation, taking x and y inside and multiplying respectively with (y-x) we get,
$(x-y)(y-x)=xy-xx-yy+yx$

Now substituting xx as x2 and yy as y2 we have,
$(x-y)(y-x)=xy-{{x}^{2}}-{{y}^{2}}+yx$
Replacing xy by yx in the above equation we have,
$(x-y)(y-x)=xy-{{x}^{2}}-{{y}^{2}}+xy$

Making necessary rearrangements in the above obtained equation we have,
$(x-y)(y-x)=xy+xy-{{x}^{2}}-{{y}^{2}}$

Adding xy to xy and making it equal to 2xy we have,
$(x-y)(y-x)=2xy-{{x}^{2}}-{{y}^{2}}$

Hence, we obtain $(x-y)(y-x)=2xy-{{x}^{2}}-{{y}^{2}}$, which is the required solution of the question.

Matching from the options given in the question we have option (c) as the correct option.

Note: The possibility of error in the question is taking wrong signs in common while multiplying x and y with (y-x) or (x-y). If we take calculation errors in the signs of x and y then it will lead to cancellation of certain terms in the expression and will ultimately give wrong solutions as a result.