Circles
Circle
star_batch_jee_advanced_2025
Grade 11
Question:
Let $P(a, b)$ be a variable point satisfying $4 \leq a^2 + b^2 \leq 9$ and $b^2 - 4ab + a^2 \leq 0$. Let $R$ be the complete region represented in $x-y$ plane in which $P$ can lie, if $m$ be the minimum value of $|a + b|$ for all position of $P$ lying in region $R$. Then $[m]$ is ___. (Where $[.]$ represents G.I.F.)
Step-by-Step Solution
Key Concept: Convert the inequality constraint to an angular range and integrate in polar coordinates.
The region $4 ≤ a^2 + b^2 ≤ 9$ represents the annulus between concentric circles of radii 2 and 3. For $b^2 - 4ab + a^2 ≤ 0$, let $\frac{b}{a} = \tan θ$, which gives $\tan θ ∈ [2 - \sqrt{3}, 2 + \sqrt{3}]$, so $θ ∈ [15°, 75°]$. The area integral becomes $\int_{15°}^{75°} \int_2^3 r \, dr \, dθ = \frac{5}{2} · 60° = \frac{5π}{6}$. The minimum chord length is $2\sqrt{2} · \frac{\sqrt{3}}{2} = \sqrt{6}$.
Correct Answer: 2