To solve this problem, let's denote the length of the circular track as $L$ meters, the speed of Brinda as $v_B$ meters per second, and the speed of Shama as $v_S$ meters per second.

Step 1: First Meeting Point

Brinda and Shama start from diametrically opposite points and run towards each other. When they first meet, Brinda has run 100 meters. Since they start from diametrically opposite points, the total distance covered by both is half the track's length: $\text{Distance covered by Brinda} + \text{Distance covered by Shama} = \frac{L}{2}$ Thus, the distance covered by Shama is: $\text{Distance covered by Shama} = \frac{L}{2} - 100 \quad \text{(Equation 1)}$

Step 2: Second Meeting Point

For their second meeting, they run towards each other again. If Shama has run 150 meters past their first meeting point, the total distance Shama has covered by the time they meet for the second time is: $\text{Distance covered by Shama} = \frac{L}{2} - 100 + 150 = \frac{L}{2} + 50$ In this case, the total distance covered by both is the full track's length: $\text{Distance covered by Brinda} + \text{Distance covered by Shama} = L$ Thus, the distance covered by Brinda during the second meeting is: $\text{Distance covered by Brinda} = L - \left(\frac{L}{2} + 50\right) = \frac{L}{2} - 50 \quad \text{(Equation 2)}$

Step 3: Speed Ratios

The ratios of the distances covered by Brinda and Shama between the first and second meeting should be consistent because both run at constant speeds. From Equation 1 and Equation 2: $\frac{100}{\frac{L}{2} - 50} = \frac{\frac{L}{2} - 100}{150}$

Step 4: Solve for $L$

We can solve this equation to find the value of $L$. Start by cross-multiplying: $100 \times 150 = \left(\frac{L}{2} - 50\right) \times \left(\frac{L}{2} - 100\right)$ This simplifies to: $15000 = \left(\frac{L}{2} - 50\right)\left(\frac{L}{2} - 100\right)$ Expand the product on the right side: $15000 = \frac{L^2}{4} - 75L + 5000$ Multiply the entire equation by 4 to eliminate the fraction: $60000 = L^2 - 300L + 20000$ Bring all terms to one side: $L^2 - 300L - 40000 = 0$

Step 5: Solve the Quadratic Equation

We can solve this quadratic equation using the quadratic formula: $L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Where $a = 1$, $b = -300$, and $c = -40000$: $L = \frac{300 \pm \sqrt{90000 + 160000}}{2}$ $L = \frac{300 \pm \sqrt{250000}}{2}$ $L = \frac{300 \pm 500}{2}$ This gives us two possible solutions: $L = \frac{800}{2} = 400 \quad \text{or} \quad L = \frac{-200}{2} = -100 \quad (\text{not feasible since length cannot be negative})$ Thus, $L = 400$ meters.

Step 6: Sum of the Digits of the Track Length

The length of the track is 400 meters, and the sum of its digits is: $4 + 0 + 0 = 4$

Therefore, the sum of the digits of the length of the track is $\boxed{4}$.

---

Would you like any further details or have any questions?

Related Questions:

1. If Brinda had run 150 meters instead of 100 meters, what would the length of the track be?
2. How would the solution change if they started from the same point instead of diametrically opposite points?
3. What is the ratio of Brinda's speed to Shama's speed?
4. If the track was 600 meters long, how long would it take for their first and second meetings?
5. How would the time intervals between meetings change if one runner was twice as fast as the other?

Tip: When solving problems involving relative motion, breaking the problem down into individual segments and equating distances or speeds often simplifies the analysis.