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Math 261 Conic Sections Handout
A Guide to Conic Sections
A. Parabolas Geometric Definition: A parabola is the set of all points in a plane equidistant from a fixed point F (the focus) and a fixed line ℓ (the directrix) in the plane.
- The axis of a parabola is the line through F perpendicular to the directrix ℓ.
- The vertex of a parabola if the point V on the axis which is halfway between the focus F and the line ℓ.
p {
F:(0,p)
V:(0,0) directrix: y = − p
P:(x,y)
P’:(x, − p)
d d
Some Useful Formulas: If V : (0, 0)
- The general form of such a parabola is: y = ax^2 or x = ay^2.
- Up/Down parabolas have equation: x^2 = 4py or y = (^41) p x^2
- Left/Right parabolas have equation: y^2 = 4px or x = (^41) p y^2
If V : (h, k)
- The general form of such a parabola is: y = ax^2 + bx + c or x = ay^2 + by + c.
- Up/Down parabolas have equation: (x − h)^2 = 4p(y − k)
- Left/Right parabolas have equation: (y − k)^2 = 4p(x − h)
For any parabola, p = (^41) a. For an Up/Down parabola, h = − 2 ba and the axis has equation x = − 2 ba. For a Left/Right parabola, k = − 2 ba and the axis has equation y = − 2 ba .. Finally, if we consider a parabolic mirror, the focus F of a parabola has interesting properties:
- If a “light source” is placed at F , then all light rays emitted will be reflected so as to travel perpendicular to the axis of the parabola.
- Similarly, a beam of light coming toward a parabolic mirror traveling perpendicular to the axis will be reflected into the focus.
B. Ellipses Geometric Definition: An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points F and F ′^ (the foci) in the plane is a positive constant. We can think of constructing an ellipse as follows: Push two thumbtacks in a sheet of paper sitting on top of some cardboard. Then take a piece of string and tie each end onto one of the tacks (with a little bit of slack left over). Finally, take a pencil, pull the string taught around the pencil and trace out a path around the two tacks, guided by the string.
- The midpoint of the line segment connecting the foci is the center of the ellipse.
- The points V and V ′^ on the ellipse that are on the line determined by F and F ′^ are called the vertices of the ellipse.
- The line segment V V ′^ is the major axis of the ellipse.
- We use M and M ′^ to denote the points on the ellipse that are on the line which is perpendicular to the line determined by F and F ′.
- The line segment M M ′^ is the minor axis of the ellipse.
- The length of the major axis is denoted by 2a, and the length of the minor axis is denoted by 2b.
C
P:(x,y)
V’(−a,0)
M(0,b)
M’(0,−b)
F’(−c,0) F(c,0) V(a,0)
Some Useful Formulas: If C : (0, 0)
- If the major axis is horizontal, the equation of an ellipse has the form: x 2 a^2 +^
y^2 b^2 = 1, and the ellipse has vertices (±a, 0), minor axis endpoints (0, ±b), and foci (±c, 0), where c^2 = a^2 − b^2.
- If the major axis is vertical, the equation of an ellipse has the form: x 2 b^2 +^
y^2 a^2 = 1, and the ellipse has vertices (0, ±a), minor axis endpoints (±b, 0), and foci (0, ±c), where c^2 = a^2 − b^2.
- The eccentricity e of an ellipse is given by e = (^) ac =
√a (^2) −b 2 a.^ Notice that 0^ < e <^ 1 for any ellipse.^ The eccentricity can be thought of as a measure of how close an ellipse is to being circular. If e ≈ 0 then the ellipse is nearly circular, while if e ≈ 1, then the ellipse is almost “flat”.
- The “reflective property” of ellipses is that if a wave or ray of light emanates from one focus of an ellipse, it will pass through the other focus.
If C : (h, k)
- If the major axis is horizontal, the equation of an ellipse has the form: (x−h)
2 a^2 +^
(y−k)^2 b^2 = 1, and the ellipse has vertices (h ± a, k), minor axis endpoints (h, k ± b), and foci (h ± c, 0), where c^2 = a^2 − b^2.
- If the major axis is vertical, the equation of an ellipse has the form: (x−h)
2 b^2 +^
(y−k)^2 a^2 = 1, and the ellipse has vertices (h, k ± a), minor axis endpoints (h ± b, k), and foci (h, k ± c), where c^2 = a^2 − b^2.
