If the curve
Question:

If the curve $y=a x^{2}+b x+c, x \in R$, passes through the point $(1,2)$ and the tangent line to this curve at origin is $y=x$, then the possible values of $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are :

1. (1) $a=1, b=1, c=0$

2. (2) $a=-1, b=1, c=1$

3. (3) $a=1, b=0, c=1$

4. (4) $a=\frac{1}{2}, b=\frac{1}{2}, c=1$

Correct Option: 1

Solution:

$2=a+b+c \ldots \ldots(i)$

$\frac{d y}{d x}=2 a x+\left.b \Rightarrow \frac{d y}{d x}\right|_{(0,0)}=1$

$\Rightarrow b=1 \Rightarrow a+c=1$

$(0,0)$ lie on curve

$\therefore \mathrm{c}=0, \mathrm{a}=1$