Find the values of a and b if the slope

Solution:

Given :

$x y+a x+b y=2 \ldots$ .....(1)

On differentiating both sides w.r.t. $x$, we get

$x \frac{d y}{d x}+y+a+b \frac{d y}{d x}=0$

$\Rightarrow \frac{d y}{d x}(x+b)=-a-y$

$\Rightarrow \frac{d y}{d x}=\frac{-a-y}{x+b}$

Now,

$\left(\frac{d y}{d x}\right)_{(1,1)}=2$

$\Rightarrow \frac{-a-1}{1+b}=2$

$\Rightarrow-a-1=2+2 b$

$\Rightarrow-a=3+2 b$

$\Rightarrow a=-(3+2 b)$

On substituting $a=-(3+2 b), \quad x=1$ and $y=1$ in eq. $(1)$, we get

$1-(3+2 b)+b=2$

$\Rightarrow 1-3-2 b+b=2$

$\Rightarrow b=-4$

and

$a=-(3+2 b)=-(3-8)=5$

$\therefore a=5$ and $b=-4$