Mathematics - Application of Derivative Question with Solution | TestHub

$f(x)=x^2-ax+b$. One root must be 2, so $f(2)=0\Rightarrow 2a-b=4\dots$ (1) ${\sum }_{i=1}^{5}f(i)=20\Rightarrow 55-15a+5b=20\Rightarrow 3a-b=7\dots (2)$

From (1) and (2) we get $a=3,b=2\therefore f(x)=x^2-3x+2$ Curve $C_1:x^2-y^2-3x+2=0$ : rectangular hyperbola with centre $\left(\frac{3}{2},0\right)$ and $a^2=b^2=\frac{1}{4}$ Curve $C_2:x^2+y^2-3x+2=0$ : circle with centre $\left(\frac{3}{2},0\right)$ and radius $\frac{1}{2}$ Normal to $C_1$ at $\left(x_0,y_0\right)$ passes through $\left(\frac{3}{2},0\right)$ only when $y_0=0$, giving the points $(1,0)$ and (2,0), at both of which the normal is the x-axis itself. Exactly one normal of $C_1$ passes through centre of $C_2$