Exercises about ellipses as conic sections
- In which of the figures below is the intersection of the pictured cone
and plane an ellipse?
solution
- If each focus is three units from the center, what is the difference
in distances between the coneÕs vertex and B and A?
solution
- Describe all points that could be vertices of a cone whose intersection
with a plane is an ellipse that is in that plane and the distance between
whose center and either focus is c = 3 units.
solution
- Describe all points that could be vertices of a cone whose intersection
with a plane is an ellipse whose rectangular
coordinate equation is
solution
Solutions
- In figures a and c the intersection of the
given cone and plane is an ellipse, because in those figures the plane
intersects all generating lines of the cone. In figure b the intersection
is a parabola, and in figure d it's a hyperbola.
-
The difference in distances between the coneÕs vertex
and A and B is the same as the difference in distances between a focus and
A and B. If the distance from that focus to the nearer of A and B is u,
then the focus is u + 6 from the other of A and B, so the difference of
these distances is (u + 6) Ð u = 6. In general, the vertex can be thought
of as being chosen so that the difference of distances to A and B is twice
the distance of a focus from the center.
-
We want a cone whose intersection with a plane is
an ellipse that is in that plane and the distance between whose center and
either focus is c = 3 units. According to the proof
of the conic section property for an ellipse, potential vertices for this
cone lie on a plane perpendicular to the plane of the ellipse and through
the foci of the ellipse, and satisfy the condition that the difference of
distances between the potential vertex and the foci of the ellipse is 2c = 2(3) = 6 units.
- In the ellipse with equation
