If f(x+2y,x,x-2y)=xy, then f(x,y) equals

Question

If f(x+2y,x,x-2y)=xy, then f(x,y) equals (A)  (x2-y2/8)  (B)  (x2-y2/4)  (C)  (x2+y2/4)  (D)  (x2-y2/2)

Tardigrade Admin · Accepted Answer

We have, f(x+2y,x-2y)=xy ... (i) Let x+2y=u and x-2y=v Then, x=(u+v/2) and y=(u-v/4) Substituting the values of x and y in Eq. (i), we obtain f(u,v)=(u2-v2/8)⇒ f(x,y)=(x2-y2/8)

Q. If $f (x + 2 y, x, x - 2 y) = x y,$ then $f (x, y)$ equals

$\frac{x ^{2} - y ^{2}}{8}$

$\frac{x ^{2} - y ^{2}}{4}$

$\frac{x ^{2} + y ^{2}}{4}$

$\frac{x ^{2} - y ^{2}}{2}$

We have, $f (x + 2 y, x - 2 y) = x y$ ... (i) Let $x + 2 y = u$ and $x - 2 y = v$ Then, $x = \frac{u + v}{2}$ and $y = \frac{u - v}{4}$ Substituting the values of $x$ and y in Eq. (i), we obtain $f (u, v) = \frac{u ^{2} - v ^{2}}{8} \Rightarrow f (x, y) = \frac{x ^{2} - y ^{2}}{8}$

Q. If f(x+2y,x,x−2y)=xy, then f(x,y) equals

8x2−y2​

4x2−y2​

4x2+y2​

2x2−y2​