Question

A pole has to be erected at a point on the boundary of a circular park of diameter 17 m in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Find the distances from the two gates where the pole is to be erected.

Answer 1

Let P be the pole to be erected and A, B be the opposite fixed gates.
Given, PA – PB = 7
Let PA = a, PB = b
Hence a – b = 7
⇒ a = 7 + b �...(1)
In right ΔPAB,
AB2 = AP2 + BP2 (By Pythagoras theorem)
132 = a2 + b2
169 = (7 + b)2 + b2
169 = 49 + 14b + 2b2
2b2 + 14b – 120 = 0
b2 + 7b – 60 = 0
b2 + 12b - 5b – 60 = 0
b(b + 12) - 5(b + 12) = 0
(b + 12)(b – 5) = 0
Hence b = 5 or – 12
But ‘b’ cannot be negative
Therefore, b = 5
⇒ a = 7 + 5 = 12
Hence, PA = 12 m and PB = 5 m

Answer 2

on the boundary of the circular park are on. ... meters in such a way that the difference of its distances from two diametrical opposite fixed gates A and B on the boundary is 7 meters.