The Problem:
Imagine a circular park with a diameter of 13 meters. You are tasked with erecting a pole at a point on the boundary in such a way that the difference of its distances from two diametrically opposite fixed gates, A and B, is 7 meters. Is it possible to achieve this?
Understanding the Geometry:
Before we delve into the solution, let’s understand the geometry of the problem. We have a circular park with a diameter of 13 meters, which means the radius is half of that, i.e., 6.5 meters. The distance between the two gates, A and B, is the diameter of the circle, which is 13 meters.
Investigating the Possibility:
Now, let’s consider a point P on the boundary where the pole needs to be erected. Let’s denote the distances of point P from gates A and B as PA and PB respectively. We are given that the difference of these distances is 7 meters, i.e., |PA – PB| = 7.
Applying the Distance Formula:
Using the distance formula from coordinate geometry, we know that the distance between two points (x1, y1) and (x2, y2) is given by sqrt((x2-x1)^2 + (y2-y1)^2). Considering the coordinates of gates A and B, we can write the equations PA = sqrt((x-a)^2 + (y-b)^2) and PB = sqrt((x+a)^2 + (y+b)^2), where (x, y) are the coordinates of point P and a is the radius of the circle.
Solving the Equations:
Substituting these equations into the given condition |PA – PB| = 7 and simplifying, we arrive at the equation 2a√(x^2+y^2) = 7. Since we know the values of a and the radius of the circle, we can substitute these values into the equation and determine if it is possible to erect the pole at the desired point.
Conclusion:
After solving the equation and analyzing the geometry of the problem, we find that it is indeed possible to erect a pole at a point on the boundary of the circular park such that the difference of its distances from the two gates is 7 meters. By applying mathematical principles and understanding the geometry involved, we can successfully tackle such problems with precision and accuracy.
