The function $f: C \rightarrow C$ defined by $f(x)=\frac{a x+d}{c x+d}$ for $x \in C$ where $b d \neq 0$ reduces to a constant function, if

• A

  $a=c$
• B

  $b=d$
• Correct answer
  C

  $a d=b e$
• D

  $a b=c d$

Solution

Now, take option (c) i.e., $a d=b c$ $\Rightarrow \quad \frac{a}{b}=\frac{c}{d}=k$ (say) From equation (i), $\begin{aligned} f(x) & =\frac{b k x+b}{d k x+d} \\ & =\frac{b(k x+1)}{d(k x+1)}=\frac{b}{d}=\text { constant } \end{aligned}$

Hence, the correct answer is option: 'C'.