Focal Points Unveiled?

📌 CONCEPT: A conic section is formed when a plane intersects a double cone, and in this case, the conic section is formed by the water surface of the inverted cone.

📐 RULE / FORMULA: The distance between the two focal points of a conic section is given by the formula: 2c, where c is the distance from the vertex of the cone to the point where the intersecting plane meets the cone.

💡 WORKED EXAMPLE: Consider a right circular cone with a base radius of 4m and a slant height of 10m. When the cone is filled with water up to 75% of its height, the conic section formed by the water surface has a distance between the two focal points of 2c, where c = 0.75 × 10m = 7.5m. Therefore, the distance is 2 × 7.5m = 15m.

⚠️ COMMON MISTAKE: Students often forget to calculate the actual distance c, and instead use the full height of the cone as the value of c. This leads to an incorrect calculation of the distance between the two focal points.