Question

Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2.

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ xy+ax+by=2...\left(1\right) \\ \text{On differentiating both sides w.r.t.}x,\mathrm{we}\mathrm{get} \\ x\frac{dy}{dx}+y+a+b\frac{dy}{dx}=0 \\ \Rightarrow \frac{dy}{dx}\left(x+b\right)=-a-y \\ \Rightarrow \frac{dy}{dx}\text{=}\frac{-a-y}{x+b} \\ \mathrm{Now}, \\ {\left(\frac{dy}{dx}\right)}_{\left(1,1\right)}=2 \\ \Rightarrow \frac{-a-1}{1+b}=2 \\ \Rightarrow -a-1=2+2b \\ \Rightarrow -a=3+2b \\ \Rightarrow a=-\left(3+2b\right) \\ \text{On substituting}a=-\left(3+2b\right),\mathit{\text{x}}\text{=1 and}\mathit{\text{y}}\text{=1 in eq. (1), we get} \\ 1-\left(3+2b\right)+b=2 \\ \Rightarrow 1-3-2b+b=2 \\ \Rightarrow b=-4 \\ \text{and} \\ a=-\left(3+2b\right)=-\left(3-8\right)=5 \\ \text{∴}a=5\text{and}b=-4 \end{gathered}$$