Question

The curve $y=a{x}^2+bx+c$ passes through the point $(1,2)$ and its tangent at origin is the line $y=x$. The area bounded by the curve, the ordinate of the curve at minima and the tangent line is

• A. $\frac{1}{24}$
• B. $\frac{1}{12}$
• C. $\frac{1}{8}$
• D. $\frac{1}{6}$

Answer 1

Answer: (A)

$x=1;y=2$
$2=a+b+c$....(1)
$x=0,y=0\Rightarrow c=0\Rightarrow a+b=2$
$\text{ now }{\frac{dy}{dx}|}_{(0,0)}=2ax+b=1$
$\therefore b=1;a=1$
$\text{ now }{\frac{dy}{dx}|}_{(0,0)}=2ax+b=1$
Hence the curve is $y=x^2+x$
$A=\underset{-\frac{1}{2}}{\overset{0}{\int }}\left(x^2+x-x\right)dx=\underset{-\frac{1}{2}}{\overset{0}{\int }}\left(x^2\right)dx=\frac{1}{24}\text{ sq. units }$