Question

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

पर्याय

• b = 1, c = 2

• b = −1, c = 1

• b = 2, c = 1

• b = −2, c = 1

Answer 1

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,

$$y=x^2+bx+c...\left(1\right)$$

$$\Rightarrow \frac{dy}{dx}=2x+b$$

$$\text{ Slope of the tangent }={\left(\frac{dy}{dx}\right)}_{(1,1)}=2+b$$

$$\text{ Given}:$$

$$\text{ Slope of the tangent }=1$$

$$\Rightarrow 2+b=1$$

$$\Rightarrow b=-1$$

$$\text{ On substituting b= - 1, x=1 and y=1 in (1), we get}$$

$$\Rightarrow 1=1-1+c$$

$$\Rightarrow c=1$$