Question

If the lines $x+y=a$ and $x-y=b$ touch the curve $y=x^2-3x+2$ at the points where the curve intersects the $x$-axis, then $\frac{a}{b}$ is equal to

Answer 1

$y=x^2-3x+2$
$\Rightarrow y=(x-1)(x-2)$
Touches $x$-axis at $(1,0)$ and $(2,0)$
Slope of lines $x+y=a$ and $x-y=b$ are $-1$ and $1.$
So, the equation 1 is
$y-0=-1(x-1)$
$\Rightarrow x+y=1\Rightarrow a=1\cdots \left(i\right)$
So, the equation 2 is
$y-0=x-2$
$\Rightarrow x-y=2\Rightarrow b=2\cdots \left(ii\right)$
Equation $\left(i\right)/\left(ii\right)$
$\frac{a}{b}=\frac{1}{2}$