Mathematics - Application of Derivative Question with Solution | TestHub

Given curve is $y=f(x)=x^2+bx-b$ $\Rightarrow {\mathrm{f}}^{\prime }(\mathrm{x})=2\mathrm{x}+\mathrm{b}$ The equation of tangent at point $(1,1)$ is $y-1={\left(\frac{dy}{dx}\right)}_{(1,1)}(x-1)$ $\Rightarrow y-1=(b+2)(x-1)$ $\Rightarrow (2+b)x-y=1+b$ $\Rightarrow \frac{x}{\left(\frac{1+b}{2+b}\right)}-\frac{y}{(1+b)}=1$
So, $\mathrm{OA}=\frac{1+\mathrm{b}}{2+\mathrm{b}}$ and $\mathrm{OB}=-(1+\mathrm{b})$ Now, area of $\Delta \mathrm{AOB}=\frac{1}{2}\times$ $\frac{(1+b)[-(1+b)]}{(2+b)}=2$ $\Rightarrow 4(2+b)+(1+b{)}^2=0$ $\Rightarrow b^2+6b+9=0$ $\Rightarrow (b+3{)}^2\Rightarrow b=-3$