Exercises about constant sum property
- An ellipse is formed by cutting a cylinder with a plane. Spheres are inscribed in the cylinder on each side of the ellipse and tangent to it. The distance along the cylinder's edge from one endpoint of the ellipse's major axis to the circle of tangency of one sphere is 3 units, and the distance from that point to the circle of tangency of the other sphere is 6 units. What's the sum of the distances from any point on the ellipse to the two foci?
solution
- An ellipse is formed by cutting a cone with a plane. Spheres are inscribed in the cone on each side of the ellipse and tangent to it. The distance along the cone's edge from one endpoint of the ellipse's major axis to the circle of tangency of one sphere is 2 units, and the distance from that point to the circle of tangency of the other sphere is 9 units. What's the sum of the distances from any point on the ellipse to the two foci?
solution
- A ovular figure is drawn by tacking two ends of a piece of string to a piece of cardboard with some slack, putting the end of a pencil inside the string and pulling tight while moving the pencil.
solution
Solutions
- The sum of the distances from any point on an ellipse to the two foci is the sum of the distances from that point along the edge of the cylinder to the circles of tangency between the inscribed spheres and the cylinder. If a point is 3 units from one circle of tangency and 6 units from another, then the constant sum for the ellipse is 3 + 6 = 9 units.
- Similarly, the sum of the distances from any point on an ellipse to the two foci is the sum of the distances from that point along the edge of the cone to the circles of tangency between the inscribed spheres and the cone. If a point is 2 units from one circle of tangency and 9 units from another, then the constant sum for the ellipse is 2 + 9 = 11 units.
- A figure drawn with a pencil in a loop of string anchored by two tacks is an ellipse because the sum of the distances from the pencil point to the two anchors is constant.
- Similarly, the sum of the distances from any point on an ellipse to the two foci is the sum of the distances from that point along the edge of the cone to the circles of tangency between the inscribed spheres and the cone. If a point is 2 units from one circle of tangency and 9 units from another, then the constant sum for the ellipse is 2 + 9 = 11 units.
