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

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

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

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x^{2}

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y^{2}

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Solution

The correct option is B $\frac{x^{2} - y^{2}}{4}$
Given $f (x + a y, x - a y) = a x y$

Let $x + a y = X \dots (1)$

$x - a y = Y \dots (2)$

$(1) + (2)$

$⟹ x + a y + x - a y = X + Y$

$⟹ 2 x = X + Y$

$⟹ x = \frac{X + Y}{2}$

$(1) - (2)$

$x + a y - x + a y = X - Y$

$2 a y = X - Y$

$⟹ y = \frac{X - Y}{2 a}$

Consider $f (x + a y, x - a y) = a x y$

Substituting The values of $x, y$

$⟹ f (X, Y) = a (\frac{X + Y}{2}) (\frac{X - Y}{2 a})$

$⟹ f (X, Y) = \frac{X^{2} - Y^{2}}{4}$

$⟹ f (x, y) = \frac{x^{2} - y^{2}}{2}$

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If f(x+ay,x−ay)=axy, then f(x,y) equals.

The correct option is B x2−y24Given f(x+ay,x−ay)=axyLet x+ay=X⋯(1)x−ay=Y⋯(2)(1)+(2)⟹x+ay+x−ay=X+Y⟹2x=X+Y⟹x=X+Y2(1)−(2)x+ay−x+ay=X−Y2ay=X−Y⟹y=X−Y2aConsider f(x+ay,x−ay)=axySubstituting The values of x,y⟹f(X,Y)=a(X+Y2)(X−Y2a)⟹f(X,Y)=X2−Y24⟹f(x,y)=x2−y22

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