Question

If the line y = x touches the curve y = x² + bx + c at a point (1, 1) then

(a) b = 1, c = 2
(b) b = −1, c = 1
(c) b = 2, c = 1
(d) b = −2, c = 1

Answer 1

(b) b = −1, c = 1

We can find the slope of the line by differentiating w.r.t. x.
Slope of the given line = 1
Now,
$$\begin{gathered} y=x^2+bx+c...\left(1\right) \\ \Rightarrow \frac{dy}{dx}=2x+b \\ \text{Slope of the tangent=}{\left(\frac{dy}{dx}\right)}_{\left(1,1\right)}\text{=2+}\mathit{\text{b}} \\ \text{Given:} \\ \text{Slope of the tangent = 1} \\ \Rightarrow 2+b=1 \\ \Rightarrow \mathrm{b}=-1 \\ \mathrm{On}\mathrm{s}\text{ubstituting}\mathit{\text{b}}\text{=}-1,\mathit{\text{x}}\text{=1 and}\mathit{\text{y}}\text{=1 in (1), we get} \\ \Rightarrow 1=1-1+c \\ \Rightarrow c=1 \end{gathered}$$