Question: A pole has to be erected at a point on the boundary of a circular park of diameter 13 meter in such a way that difference of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meter. Is it possible? If yes, at what distances from the two gates should the pole be?
Approach 1: let P be pole and A and B gates on either sides. Joining AB forms a semicircle thus angle APB = 90deg. By setting up a quadratic and solving
Approach 2: let P be pole and A and B gates on either sides. Given diameter = 13m curved length of semicircle is 6.5π meter now assuming arc(AP) > arc(PB) setting up a system of L.E. and solving them
x - y = 7
x + y = 6.5π
Which method was implied by the question or are both equivalent
I was typing a long Comment , which exceeded the Character limit , hence Posting this Answer , to aid OP.
[[A]] Interpretations 1 & 2 : Both are Possible , though "Straight line Distance" is very Common & Implicit , unless the "Curved Distance" is Explicitly given.
[[B]] Here are my "Equivalent Questions" which will show that Straight line or Curve may make more sense :
(B1) "Instead of Pole , think you have to stand there & throw a Ball. One Gate requires 7m more than the Other"
(B2) "Instead of Circular race , you are in a Circular merry-go-round. One Partner is much ahead of you. The Other Partner is much behind you & 7m more. You have to throw a toy to Both"
In these 2 Cases , will you use the Curve (Arc) or the Straight line (Chord) to measure along ? Straight makes more sense.
(B3) You have to make Wooden wall & Metal wall around the Park. Wooden wall is 7m more than Metal wall.
(B4) You have to Paint the wall around the Park. White wall is 7 times the Green wall.
In these 2 Cases , will you use the Curve (Arc) or the Straight line (Chord) to measure along ? Curve makes more sense.
[[C]] Is Straight line (Chord) Solution Equivalent to Curve (Arc) Solution ?
Certain Cases it will be Equivalent , though not here.
Eg : When we have Distance is "one-seventh" less , or 7 times more ( Proportionally less or Proportionally more ) , then : Both are Equivalent.
When we have Distance is less by 7m or more by 7m ( Absolutely less or Absolutely more ) , it is not Proportionally varying , hence Both are not Equivalent.
In other words , Proportionally varying Case is when we can Zoom-In or Zoom-Out (Scaling) the Circle & then the ratios will be Constant. Arc length & Chord length may be used Equivalently & Interchangeably.
Absolute Distance will vary & will not be Equivalent.
[[D]] Solving your Question :
Method 1 (which is Common & which your teacher is asking) :
$x^2+(x+7)^2=13^2$ , which gives $2x^2+14x+49=169$
$x^2+7x-60=0$
$x \approx 4.97$
Scale the Circle to Double Size , we will get :
$x^2+(x+7)^2=26^2$ & $x \approx 4.97$ will not work , because 7 is Absolute Change , not Proportional Change.
Method 2 (which is valid, though not which your teacher is asking) :
Semi-Circle Arc length is $\pi 13 /2$
$x+(x+7)=13\pi$ which gives $2x = \pi 13 /2 - 7$
$x \approx 13.42$
Not Equivalent.