# 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

Now, Angle in a semicircle is a right angle, therefore ABP is a triangle right-angled at P,
Therefore, By Pythagoras theorem i.e.

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 = -5 or a = 12 m

but distance can't be negative, hence a =  12

Thus, distance of pole is 12 m from gate A and (12 - 7) = 5 meters from gate B

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