Question

If the slope of the curve $y=\frac{ax}{b-x}$ at the point $(1,1)$ is $2$, then the values of $a$ and $b$ are respectively

• A. $1,-2$
• B. $-1,2$
• C. $1,2$
• D. None of these

Answer 1

Answer: (C)

$1,2$

$y=\frac{ax}{b-x}$

Slope= $\frac{dy}{dx}$

So,$\frac{dy}{dx}$= $\frac{a(b-x)-(-1)\left(ax\right)}{{(b-1)}^2}⟶\left(1\right)$

for $(1,1)$

put in equation of curve-

$1=\frac{a}{(b-1)}\Rightarrow (b-1)=a⟶\left(2\right)$

From slop after putting (1,1)

We get,

$2=\frac{a(b-1)+a}{{(b-1)}^2}$

put $(b-1)=a$ from equation(2)

$2=\frac{a^2+a}{a^2}\Rightarrow 2a^2=a^2+a\Rightarrow a(a-1)=0\Rightarrow a=0,1$

put in (2),

We get

$b=1,2$

In option only one value is given

$\therefore a=1,b=2$