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 an the tangent line is

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

Answer 1

The correct option is C $\frac{1}{8}$

$y=a{x}^2+bx+c$ point $(1,2)$ $y=x$

$2=a\left(1\right)+b+c$ ....... $\left(1\right)$

$a+b+c=2$ $x=y$ $m=1$

$\frac{dy}{dx}=2ax+b$

at origin $(0,0)\frac{dy}{dx}=2a\left(0\right)+b=b$

Since $\frac{dy}{dx}=m\Rightarrow m=b=1$

put $x=0$ y=0$inequationy=a{x}^2+bx+c$

then we get $c=0$

Put $b,c$ values in equation

$2=a+b$

$a=1$

$y=x^2+k\Rightarrow y=x^2+x\Rightarrow$ minimum $\Rightarrow \frac{1}{8}$