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1.Relation and Function
hard
If $f(x + ay,\;x - ay) = axy$, then $f(x,\;y)$ is equal to
A
$xy$
B
${x^2} - {a^2}{y^2}$
C
$\frac{{{x^2} - {y^2}}}{4}$
D
$\frac{{{x^2} - {y^2}}}{{{a^2}}}$
Solution
(c) Given $f(x + ay,\,x – ay) = axy$…..$(i)$
Let $x + ay = u$ and $x – ay = v$
Then $x = \frac{{u + v}}{2}$ and $y = \frac{{u – v}}{{2a}}$
Substituting the value of $x$ and $y$ in $(i)$, we obtain
$f(u,v) = \frac{{{u^2} – {v^2}}}{4}$ ==> $f(x,\,y) = \frac{{{x^2} – {y^2}}}{4}$.
Standard 12
Mathematics