Question

The diameter of a circular garden is 13 metres. The distance between two gates of the garden is 13 metres. An electric pole is to be erected on the circumference of the garden such that the difference between the distance of the pole from the gates is 7 metres. Can such a pole be erected? If yes, find the distance of the pole from each gate.

Answer 1

AB is the diameter of the garden.

AB = 13 m.

Let A and B represent the gates of the garden.

Let P be the position of the electric pole on the circumference of the garden.

Let AP be x m.

Then from the given condition,

BP = (x + 7) m.

∠ APB = 90°      ...(Angle inscribed in a semicircle.)

∴ △APB is a right angled triangle.

By Pythagoras theorem,

AB² = AP² + BP²

∴ (13)² = (x)² + (x + 7)²

∴ x² + x² + 14x + 49 = 169         ... (Transposing the sides)

∴ 2x² + 14x + 49 − 169 = 0

∴ 2x² + 14x − 120 = 0

∴ x² + 7x − 60 = 0          ...(Dividing by 2)

∴ x² + 12x − 5x − 60 = 0

∴ x(x + 12) − 5(x + 12) = 0

∴ (x + 12)(x − 5) = 0

∴ x + 12 = 0 or x − 5 = 0

∴ x = −12 or x = 5

But the distance cannot be negative.

∴ x = −12 is unacceptable.

∴ x = 5

x + 7 = 5 + 7 = 12

∴ AP = 5 m, and BP = 12 m

Garden poles can be placed around the perimeter. The pole will be 5 m from one gate and 12 m from the other. [Four pole positions exist. 2 on the lower diameter side (P and P₁) and 2 on the upper diameter side (P₂ and P₃).]