Question

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