Question

If the slope of the curve $y=\frac{a x}{b-x}$ at the point $(1,1)$ is $2$ , then find the value of $a+b$.

Answer 1

We have, $y=\frac{a x}{b-x}$
$\Rightarrow \frac{d y}{d x}=\frac{(b-x) a-a x \cdot(-1)}{(b-x)^{2}}=\frac{a b}{(b-x)^{2}}$
$\therefore \left[\frac{d y}{d x}\right]_{(1,1)}=\frac{a b}{(b-1)^{2}}=2$ (given) .....(i)
Since the curve passes through the point $(1,1)$, therefore,
$1=\frac{a}{b-1} $
$\Rightarrow a=b-1$
On putting $a=b-1$ in equation (i), we get
$\frac{(b-1) b}{(b-1)^{2}}=2$
$\Rightarrow b=2$
$\therefore a=2-1=1$
Hence, $a=1, b=2$