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 km north of the north gate, and it can just be seen from a point 9 km east of the south gate. What is the diameter of the wall that surrounds the city?
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 km north of the north gate, and it can just be seen from a point 9 km east of the south gate. What is the diameter of the wall that surrounds the city?
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Visualising the layout and assigning coordinates simplifies circular geometry problems involving gates and tangents.
Updated On: Aug 4, 2025
- 6 km
- 9 km
- 12 km
- None of these
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The Correct Option is C
Solution and Explanation
Let $R$ be radius of the wall. North gate is at distance $R$ from centre, south gate opposite side also at $R$. House is 3 km north of north gate $\Rightarrow$ from centre distance = $R + 3$. Point 9 km east of south gate is at coordinates $(R+9, -R)$. Distance between house and observation point is tangent line to circle. Using geometry, right triangle with vertical leg $(R+3) + R = 2R+3$ and horizontal leg $R+9$. Pythagoras on tangent condition gives $R = 6$, hence diameter $= 12$ km.
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