Area Under the Curve
Area Ratio — Line Dividing Region
nta_pyq_2026_jan
Grade 12
Question:
Let the line $x=-1$ divide the area of the region $\{(x,y):1+x^2\leq y\leq3-x\}$ in the ratio $m:n$, $\gcd(m,n)=1$. Then $m+n$ is equal to
Step-by-Step Solution
Key Concept: Curves $y=1+x^2$ and $y=3-x$ intersect at $x^2+x-2=0\Rightarrow x=1,-2$. Total area $=\int_{-2}^1[(3-x)-(1+x^2)]dx$. The line $x=-1$ divides it into $x\in[-2,-1]$ and $x\in[-1,1]$.
Ratio $m:n=20:7$. $m+n=27$.
Correct Answer: 1