Question

If the line $y=x$ touches the curve $y=x^2+bx+c$ 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

The correct option is B $b=-1,c=1$

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

Given that $y=x$ touches $\left(1\right)$

when,

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

for x and y at $(1,1)$

$1=1+b+c$

$b+c=0$

$b=-c$

from $\left(2\right)$

$x=x^2-cx+c$

$\Rightarrow x^2-cx-x+c=0$

$\Rightarrow x(x-c)-1(x-c)=0$

$\Rightarrow (x-1)(x-c)=0$

$x=1,c=x$

$c=1$

when $c=1$

$b=-c$

$b=-1$.