A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the diff

Solution

Let P be the required location of the pole. Let the distance of the pole from the gate R be x m, i.e. RP = x m. Now the difference of the distances of the pole from the two gates = QP - RP (or, RP – QP) = 7 m. Therefore, QP = (x + 7) m.

Now, QR = 13m, and since QR is a diameter.

(Angle in a semicircle is right angle)

Therefore, (By Pythagoras theorem)

i.e.,

i.e.,

i.e.,

So, the distance x of the pole from gate R satisfies the equation

We will find its discriminant and find whether it would be possible  

> 0

So, the given quadratic equation has two real roots, and it is possible to erect the pole on the boundary of the park.

Solving the quadratic equation, by quadratic formula, we have

Therefore, x = 5 or –12.

Since x is the distance between the pole and the gate R, it must be positive. Therefore, x = –12 is rejected. So, x = 5.

Thus, the pole has to be erected on the boundary of the park at a distance of 5m from the gate R and 12 m from the gate Q.