Question:
If the line $y=x$ touches the curve $y=x^{2}+b x+c$ at a point $(1,1)$ then
A. $b=1, c=2$
B. $b=-1, c=1$
C. $b=2, c=1$
D. $b=-2, c=1$
Solution:
Given that line $y=x$ touches the curve $y=x^{2}+b x+c$ at a point $(1,1)$
Slope of line $=1$
Slope of tangent to the curve $=1$
$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=2 \mathrm{x}+\mathrm{b}$
$\Rightarrow 2 \mathrm{x}+\mathrm{b}=1$
$\Rightarrow 2+\mathrm{b}=1$
$\Rightarrow \mathrm{b}=-1$
Putting this and $x=1$ and $y=1$ in the equation of the curve,
$1=1-1+c$
$\Rightarrow c=1$
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