If the parabolas y = x2 + ax + b and y = x ( c - x) touch each other at the point (1, 0), then a+ b + c =

We have,

$C_{1} : y = x^{2} + a x + b$ …(i)

$C_{2} : y = c x - x^{2}$ …(ii)

$∴ {(\frac{d y}{d x})}_{C_{1}} = 2 x + a$ and ${(\frac{d y}{d x})}_{C_{2}} = c - 2 x$

At point $(1, 0),$ we have

${(\frac{d y}{d x})}_{C_{1}} = 2 + a$ and ${(\frac{d y}{d x})}_{C_{2}} = c - 2$

If the two parabolas touch each other at $(1, 0),$ then

$2 + a = c - 2 \Rightarrow a - c + 4 = 0$ …(iii)

Since $(1, 0),$ lies on the two parabolas. Therefore,

$1 + a + b = 0$ and $c - 1 = 0$

$\Rightarrow c = 1$ and $a + b = - 1$ …(iv)

Solving (iii) and (iv), we get

$a = - 3, b = 2$ and $c = 1 \Rightarrow a + b + c = 0$

If the parabolas $y = x^{2} + a x + b$ and $y = x (c - x)$ touch each other at the point $(1, 0),$ then $a + b + c =$