Area Under the Curve
Parabolas — Area Ratio Condition to Find α
nta_pyq_2026_jan
Grade 12
Question:
Let $P_1:y=4x^2$ and $P_2:y=x^2+27$ be two parabolas. If the area of the bounded region enclosed between $P_1$ and $P_2$ is six times the area of the bounded region enclosed between the line $y=\alpha x$, $\alpha>0$ and $P_1$, then $\alpha$ is equal to:
Step-by-Step Solution
Key Concept: Intersection of $P_1$ and $P_2$: $4x^2=x^2+27\Rightarrow x=\pm3$. Area $=\int_{-3}^3(x^2+27-4x^2)dx=2\int_0^3(27-3x^2)dx=2[27x-x^3]_0^3=2(81-27)=108$.
$\alpha=12$.
Correct Answer: 1