Question

A pole has to be erected at a point on the boundary of a circular park of diameter $\sf 13 meters$ in such a way that the difference of its diameter from two diametrically opposite fixed gates A and B on the boundary is $\sf 7 meters$ . What distance from the two gates should the pole be erected ?

Answer 1

Answer:

Refer to the attachment for Diagram

Let the pole be erected at point P and A, B be the points where the gates are erected.

Now, according to the question, it is given that the difference of the distances from P to each gate is 7 m.

That is, Assuming PA > PB, we get that,

PA - PB = 7 m

⇒ PA = 7 + PB

Now we know that, ∠ APB = 90°  [ Angle in a semicircle ]

So Δ PAB is a right angled triangle

⇒ PA² + PB² = AB²

⇒ ( 7 + PB )² + PB² = 13²

⇒ ( 49 + 2 × 7 × PB + PB² ) + PB² = 169

⇒ 49 + 14 PB + 2 PB² = 169

⇒ 2 PB² + 14 PB + 49 - 169 = 0

⇒ 2 PB² + 14 PB + 120 = 0

Dividing by 2 throughout the equation we get,

⇒ PB² + 7 PB + 60 = 0

⇒ PB² + 12 PB - 5 PB + 60 = 0

⇒ PB ( PB + 12 ) - 5 ( PB + 12 ) = 0

⇒ ( PB - 5 ) ( PB + 12 ) = 0

⇒ PB = 5, -12

-12 cannot be the answer as distance is always positive.

Hence PB is 5 m.

⇒ PA = 7 + PB = 7 + 5 = 12 m.

Hence the pole must be erected at a distance of 12 m from Gate A and 5 m from Gate B.