Area Under the Curve
Area Between Two Parabolas — Optimization
nta_pyq_2024_jan
Grade 12
Question:
The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2y^2=kx$ and $ky^2=2(y-x)$ is maximum, is equal to:
Step-by-Step Solution
Key Concept: Find intersection points of the two parabolas, express area as a function of $k$, then maximize. Area $A=\frac{2}{3}\cdot\frac{4}{(k+4/k)^2}$. By AM-GM, $k+4/k\ge4$ (for $k>0$), minimum when $k=4/k\Rightarrow k=\pm2$.
Area $=\frac{4}{6}\cdot\frac{4}{(k+4/k)^2}$. Minimum of $k+4/k$ at $k=\pm2$. $k^2=4$ for both. Sum of squares $=4+4=8$.
Correct Answer: 8