Question

A pole is to be erected at a point on the boundary of a circular park of diameter 17m in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary 7metes. Find the distance from the two gates where the pole is to be erected.

Answer 1

the distance of the pole from gate A is 15m and gate B is 8m

Answer 2

Refer to the diagram.
Let AB be 'l', AC be 'x' and BC be 'y'.
The key here is to know that any point on the circle (expect A and B) subtend a 90° angle when connected to the two ends of the diameter.
Here, point C denotes the pole.
According to the question,
l=17m
Also, let y>x,
Then, y-x=7m
Applying Pythagorean Theorem on ∆ABC,
${l}^{2} = {x}^{2} + {y}^{2} \\ putting \: y = 7 + x \\ {l}^{2} = {x}^{2} + {x}^{2} + 14x + 49 \\ 289 = 2 {x}^{2} + 14x + 49 \\ {x}^{2} + 7x - 120 = 0 \\ (x - 8)(x + 15) = 0 \\ x = 8 \: or \: x = - 15 \\ distance \: cannot \: be \: negative \\ x = 8. \\ thus \\ y = 7 + 8 \\ y = 15.$
Hope you find this useful.