13. Conic Sections

Question 1

Forming a Parabola

Answer 1

The plane is parallel to a lateral edge of the cone.

Question 2

Forming a Hyperbola

Answer 2

The plane is parallel to the axis of the cone.

Question 3

Forming an Ellipse

Answer 3

The plane is neither parallel to a lateral edge nor the axis of the cone.

Question 4

Definition of a parabola

Answer 4

The set of points that are equidistance from a given point (focus) and a given line (directrix). The center of a parabola is its vertex.

Question 5

If a parabola has an x-orientation…

Answer 5

• It opens to the left (p<0) or right (p>0)
• The standard equation is (y–k)² = 4p(x–h)
• The focus is (h+p, k), and the directrix is x = h–p
• The vertex is (h, k)
• The common distance from the parabola to the focus and directrix is |p|

Question 6

If a parabola has an y-orientation…

Answer 6

• It opens up (p>0) or down (p<0)
• The standard equation is (x–h)² = 4p(y–k)
• The focus is (h, k+p), and the directrix is y = k–p
• The vertex is (h, k)
• The common distance from the parabola to the focus and directrix is |p|

Question 7

Definition of an ellipse

Answer 7

The set of points whose distances from two given points (foci) sum to a constant.

Question 8

If an ellipse has an x-orientation…

Answer 8

• The standard equation is (x–h)²/a² + (y–k)²/b² = 1, where a² > b²
• The major axis is parallel to the x-axis
• The center is (h, k)
• The vertices are (h–a, k) and (h+a, k), the endpoints of the major axis. So the major axis is 2a long.
• The endpoints of the minor axis are (h, k–b) and (h, k+b). So the minor axis is 2b long.
• The foci are on the major axis. Each focus is at a distance of c = √(a²–b²) from the center, so the foci are at (h–c, k) and (h+c, k).

Question 9

If an ellipse has an y-orientation…

Answer 9

• The standard equation is (x–h)²/b² + (y–k)²/a² = 1, where a² > b²
• The major axis is parallel to the y-axis
• The center is (h, k)
• The vertices are (h, k–a) and (h, k+a), the endpoints of the major axis.
• The endpoints of the minor axis are (h–b, k) and (h+b, k).
• The foci are on the major axis. Each focus is at a distance of c = √(a²–b²) from the center, so the foci are at (h, k–c) and (h, k+c).

Question 10

Definition of a hyperbola

Answer 10

The set of points whose distances from two fixed points (foci) differ by a constant. A hyperbola has two halves, corresponding to the two nappes of the cone.

Question 11

If a hyperbola has an x-orientation…

Answer 11

• The hyperbola opens to the sides.
• The standard equation is (x–h)²/a² – (y–k)²/b² = 1.
• The center is (h, k).
• The vertices are (h–a, k) and (h+a, k). The segment joining the two vertices is called the transverse axis. This axis is horizontal and has length 2a.
• The foci are on the transverse axis. The distance between the center and each focus is c = √(a²+b²), so the foci are (h–c, k) and (h+c, k).
• The vertical segment through the center with endpoints (h, k–b) and (h, k+b), which has length 2b, is called the conjugate axis. The endpoints of this axis are not on the hyperbola.
• The equations of the asymptotes are y–k = ±b/a(x–h).

Question 12

If a hyperbola has a y-orientation…

Answer 12

• The hyperbola opens up and down.
• The standard equation is (y–k)²/a² – (x–h)²/b² = 1.
• The center is (h, k).
• The vertices are (h, k–a) and (h, k+a). The transverse axis is vertical and has length 2a.
• The foci are (h, k–c) and (h, k+c).
• The conjugate axis is horizontal, has endpoints (h–b, k) and (h+b, k), and has length 2b.
• The equations of the asymptotes are y–k = ±a/b(x–h).

Question 13

Eccentricity of an ellipse or hyperbola

Answer 13

The measure of its degree of elongation.

The eccentricity is c/a.

Question 14

Standard equation of a circle

Answer 14

(x–h)² + (y–k)² = r²

with center at (h, k) and radius = r