Step 1: Analyze the given data and statements. From statement I: - The ground dimensions are 80m × 50m. - This determines the positions of Raju and Ratan at the mid-points of the shorter sides (at (25, 0) and (25, 80), respectively).
From statement II: - The area of the triangle formed by Raju, Rivu, and Ratan is 1,000 square metres. - Using the formula for the area of a triangle:
Area \(= \frac{1}{2} \times \text{base} \times \text{height},\)
where the base is the distance between Raju and Ratan (80m), we can find Rivu's coordinates.
Step 2: Calculate the time taken. The speed of the ball is constant at 10m/s. Using the coordinates of Rivu, the distances between Raju, Rivu, and Ratan can be determined, and hence the total time can be calculated:
Time \(= \frac{\text{Total Distance}}{\text{Speed}} + \text{Holding Time.}\)
Both statements I and II are necessary to calculate the total distance and time.
Answer: I and II both are required to solve the problem.
The center of a circle $ C $ is at the center of the ellipse $ E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $, where $ a>b $. Let $ C $ pass through the foci $ F_1 $ and $ F_2 $ of $ E $ such that the circle $ C $ and the ellipse $ E $ intersect at four points. Let $ P $ be one of these four points. If the area of the triangle $ PF_1F_2 $ is 30 and the length of the major axis of $ E $ is 17, then the distance between the foci of $ E $ is:
A | B | C | D | Average |
---|---|---|---|---|
3 | 4 | 4 | ? | 4 |
3 | ? | 5 | ? | 4 |
? | 3 | 3 | ? | 4 |
? | ? | ? | ? | 4.25 |
4 | 4 | 4 | 4.25 |