A pole has to be erected at a point on the boundary of a circular park

1.	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?
Answer» 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

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?

Answer»

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