The diameter of circular garden is 13 meter. The distance between two gates of garden is 13 meter. An electric pole is to be erected on the circumference of the garden such that the difference between distance of the pole from each gate is 7 meter. C…

Answer: Yes, such a pole can be erected; its distances from the gates are 12 m and 5 m.

Let the gates be A and B. Since AB = 13 m equals the diameter, AB is a diameter of the garden.

Let the pole be at P on the circumference. \(\angle APB\) is an angle in a semicircle, so \(\angle APB = 90^\circ\). Hence \(PA^2 + PB^2 = AB^2 = 169\).

Given \(PA - PB = 7\), so \((PA - PB)^2 = 49\) and \(PA \cdot PB = \dfrac{169 - 49}{2} = 60\).

\((PA + PB)^2 = 169 + 2(60) = 289\), so \(PA + PB = 17\).

Solving with \(PA - PB = 7\): \(PA = 12\) m and \(PB = 5\) m. Both are positive and less than the diameter, so the pole can be erected.