Question

A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and west. A house stands outside the city, $3	ext{ km}$ north of the north gate, and it can just be seen from a point $9	ext{ km}$ east of the south gate. What is the diameter of the wall that surrounds the city?

Accepted Answer

B. Let center of circle be origin $(0,0)$ and radius be $r$. North gate $N = (0, r)$, South gate $S = (0, -r)$. House position $A = (0, r+3)$. Point of view $P = (9, -r)$. Line connecting $A(0, r+3)$ and $P(9, -r)$ is tangent to the circle $x^2+y^2=r^2$. Slope of line $AP = \frac{-r - (r+3)}{9 - 0} = \frac{-2r-3}{9}$. Using distance from origin $(0,0)$ to line $AP = r$ gives $r = 4.5	ext{ km}$. Diameter $= 2r = 9	ext{ km}$.

Circular Wall and Gate Sightline

Explanation

Practise this under exam conditions