Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
A wire $20cm$ long be divided into two parts, if one part is to be bent into a circle, the other part is to be bent into a square and the two plane figures are to have areas the sum of which is maximum, then side length of square is_____.
Step-by-Step Solution
Key Concept: Converting the discrete sum to a Riemann integral as $n \to \infty$ allows exact evaluation of the limiting arc length sum.
For points $A_k = (2t_k^2, t_k)$ on the parabola, the slope of $FA_k$ is $\frac{t_k^2 - 1}{2t_k} = \tan(2\theta_k)$, where $t_k = \tan \phi_k$. This gives $\phi_k = \frac{\theta_k}{2} - \frac{k\pi}{4n}$. The tangent at $A_k$ makes angle $\frac{\pi}{4}$ with the $x$-axis. Computing $\lim_{n \to \infty} \sum_{k=1}^n FA_k = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \sec^2\left(\frac{k\pi}{4n}\right) = \int_0^{\pi/4} \sec^2 x \, dx = \frac{4}{\pi}$. Hence $m = 4$.
Correct Answer: 0