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Question
If the line y = x touches the curve y = x2 + bx + c at a point (1, 1) then _____________ .
Options
b = 1, c = 2
b = −1, c = 1
b = 2, c = 1
b = −2, c = 1
Solution
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,
\[y = x^2 + bx + c . . . \left( 1 \right)\]
\[ \Rightarrow \frac{dy}{dx} = 2x + b\]
\[\text { Slope of the tangent }= \left( \frac{dy}{dx} \right)_\left( 1, 1 \right) =2+b\]
\[\text { Given}:\]
\[\text { Slope of the tangent } = 1\]
\[ \Rightarrow 2 + b = 1\]
\[ \Rightarrow b = - 1\]
\[\text { On substituting b= - 1, x=1 and y=1 in (1), we get}\]
\[ \Rightarrow 1 = 1 - 1 + c\]
\[ \Rightarrow c = 1\]
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