Question

A pole has to be erected at a point on the boundary of a circular park of diameter 17 metre in such a way that the difference of its distance from two diametrically opposite in fixed gates A and B on the boundary is 7 metres. Is it possible to do so? Yes, at what distance from the two gates should the pole be erected?

Answer 1

Hint- Here we will proceed by assuming the position of pole, opposite fixed gates as variables. Then we will use Pythagoras theorem i.e. ${\left( {hypotenuse} \right)^2} = {\left( {base} \right)^2} + {\left( {perpendicular} \right)^2}$ such that we get the distance from both the gates to pole.

Complete step-by-step answer:
Formula of Pythagoras theorem- ${\left( {hypotenuse} \right)^2} = {\left( {base} \right)^2} + {\left( {perpendicular} \right)^2}$

Let us assume,
P be the position of the pole, A and B be the opposite fixed gates.
Also we are given that the difference from two diametrically opposite in fixed gates is 7 metres
$ \Rightarrow $ PA-PB=7m
Or a-b=7
Or a=7+b ……………… (1)
Now in PAB,
$ \Rightarrow A{B^2} = A{P^2} + B{P^2}$
$ \Rightarrow \left( {17} \right) = {\left( a \right)^2} + \left( {{b^2}} \right)$
Or ${a^2} + {b^2} = 289$
Putting the value of a=7 + b in the above,
$ \Rightarrow {\left( {7 + b} \right)^2} + {b^2} = 289$
Or $49 + 14b + 2{b^2} = 289$
Or $2{b^2} + 14b + 49 - 289 = 0$
Or $2{b^2} + 14b - 240 = 0$
Dividing the above by 2, we get
$ \Rightarrow {b^2} + 7b - 120 = 0$
Or ${b^2} + 15b - 8b - 120 = 0$
Or $b\left( {b + 15} \right) - 8\left( {b + 15} \right) = 0$
Or $\left( {b - 8} \right)\left( {b + 15} \right) = 0$
Either b = 8 or b = -15
Since this value cannot be negative, so b = 8 is the correct value.
Now putting b = 8 in equation 1,
We get
a = 7 + 8
a = 15
Hence PA = 15m and PB = 8m
So, the distance from gate A to pole is 15m and from gate B to the pole is 8m.
Note – We can also verify this answer using the Pythagoras theorem i.e. $A{B^2} = A{P^2} + B{P^2}$where we will put the values of AB, AP and BP so that LHS=RHS. This implies that the above calculated answer is right.