Math 445: Problems Likely to Appear on Quiz 3
1. Parabolas
- State the definition of a parabola (the one in terms of distances in the
plane and the focus point and directrix line, not the intersection of a cone
with a plane).
- Draw a figure consisting of a point F and a line d. Construct a "random"
point P on the parabola with focus F and directrix d. Also construct the line
p through P which is tangent to the parabola. Write down the major steps of
your construction so that it is clear what you did and how other points would
be constructed.
- Tell why the construction works. Specifically, why is the point P on the
parabola and why does p only intersect the parabola at P?
- State the optical property of the parabola. What beams of light will focus
at F?
2. Ellipses
- State the definition of an ellipse (the one in terms of distances in the
plane and two focus points, not the intersection of a cone with a plane).
- Draw a figure consisting of 2 points F and G and a circle centered at G
with F inside the circle. Construct a "random" point P on the ellipse with
foci F and G with distance sum from the definition = the radius of the circle.
Also construct the line p through P which is tangent to the ellipse. Write
down the major steps of your construction so that it is clear what you did
and how other points would be constructed.
- Tell why the construction works. Specifically, why is the point P on the
ellipse and why does p only intersect the ellipse at P?
- State the optical property of the ellipse. What beams of light will focus
at F?
3. Hyperbolas
- State the definition of a hyperbola (the one in terms of distances in the
plane and two focus points, not the intersection of a cone with a plane).
- Draw a figure consisting of 2 points F and G and a circle centered at G
with F outside the circle. Construct a "random" point P on the hyperbola with
foci F and G with distance difference from the definition = the radius of
the circle. Also construct the line p through P which is tangent to the hyperbola.
Write down the major steps of your construction so that it is clear what you
did and how other points would be constructed.
- Tell why the construction works. Specifically, why is the point P on the
hyperbola and why does p only intersect the hyperbola at P?
- State the optical property of the hyperbola. What relates beams of light
from P to F and P to G?
4. Cubes in spheres
Suppose a cube is inscribed in a sphere of radius 1.
- What is the shortest Euclidean (3D) distance between two vertices of the
cube?
What is the spherical distance, measured in degrees or radians,
between these two vertices? (You can leave this answer in terms of an inverse
trig function.)
5. Tetrahedra in spheres
Suppose a regular tetrahedron is inscribed in a sphere of radius 1.
- What is the Euclidean (3D) distance between two vertices of the tetrahedron?
(Hint: the 4 vertices of a regular tetrahedron also are 4 of the 8 vertices
of a cube, so you can work with a cube.)
- What is the spherical distance, measured in degrees or radians, between
these two vertices? (You can leave this answer in terms of an inverse trig
function.)