A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Is it the possible to do so? If yes, at what distances from the two gates should the pole be erected?
Let the distance of pole from gate A be ‘a’.
Difference of the distance of the pole from two diametrically opposite fixed gates A and B on the boundary is 7 metres.
Distance of pole from gate B = a – 7 m
Diameter of the park = 13 m
Hypotenuse2 = length2 + breadth2
⇒ 132 = a2 + (a – 7)2
⇒ 169 = 2a2 + 49 – 14a
⇒ a2 – 7a – 60 = 0
⇒ a2 – 12a + 5a – 60 = 0
⇒ a(a – 12) + 5(a – 12) = 0
⇒ (a + 5)(a – 12) = 0
⇒ a = 12 m
Thus distance of pole is 12 m from gate A and 5 m from gate B