Problem: The expression P+QP−Q−P−QP+Q\dfrac{P+Q}{P-Q}-\dfrac{P-Q}{P+Q}P−QP+Q−P+QP−Q, where P=x+yP=x+yP=x+y and Q=x−yQ=x-yQ=x−y, is equivalent to:
Answer Choices:
A. x2−y2xy\dfrac{x^{2}-y^{2}}{x y}xyx2−y2
B. x2−y22xy\dfrac{x^{2}-y^{2}}{2 x y}2xyx2−y2
C. 111
D. x4+y2xy\dfrac{x^{4}+y^{2}}{x y}xyx4+y2
E. x2+y22xy\dfrac{x^{2}+y^{2}}{2 x y}2xyx2+y2 Solution:
Since P=x+yP = x+yP=x+y and Q=x−y,P+Q=2x,P−Q=2y.∴P+QP−Q−P−QP+Q=2x2y−2y2x=x2−y2xyQ=x-y, P+Q=2 x, P-Q=2 y . \quad \therefore \dfrac{P+Q}{P-Q}-\dfrac{P-Q}{P+Q}=\dfrac{2 x}{2 y}-\dfrac{2 y}{2 x}=\dfrac{x^{2}-y^{2}}{x y}Q=x−y,P+Q=2x,P−Q=2y.∴P−QP+Q−P+QP−Q=2y2x−2x2y=xyx2−y2 \quad