Conic Sections

Question 1

What is the equation for a circle?

Answer 1

(x – h)² + (y – k)² = r²

Question 2

What is a parabola (in conics)?

Answer 2

A set of points where each point is equidistant from the focus and the directrix.

Question 3

What is the equation of a parabola in the x-orientation?

Answer 3

y² = 4px, focus at (p,0), and directrix at x = -p. Opens right when p>0 and left when p<0.

Question 4

What is the equation of a parabola in the y-orientation?

Answer 4

x² = 4py, focus at (0,p), and directrix at y = -p. Opens up when p>0 and down when p<0.

Question 5

What is the equation of an ellipse in the x-orientation?

Answer 5

(x – h)² (y – k)²
———– + ———– = 1, major axis length = 2a, minor axis
a² b² length 2b, foci at (c,0) & (-c,0)
where c² = a² - b² and a² > b²

Question 6

What is the equation of an ellipse in the y-orientation?

Answer 6

(x – h)² (y – k)²
———– + ———– = 1, major axis length = 2b, minor axis
b² a² length 2a, foci at (0,c) & (0,-c)
where c² = a² - b² and a² > b²

Question 7

What is the equation of a hyperbola in the x-orientation?

Answer 7

(x – h)² (y – k)²
———– - ———– = 1, opens left and right, vertices at (-a,0)
a² b² & (a,0), asymtote equations y = ±bx/a,
transverse axis length = 2a, foci at
(c,0) & (-c,0) where c² = a² - b².

Question 8

What is the equation of a hyperbola in the y-orientation?

Answer 8

(y – k)² (x – h)²
———– - ———– = 1, opens up and down, vertices at (0,-a)
b² a² & (0,a), asymtote equations y = ±ax/b,
transverse axis length = 2a, foci at
(0,c) & (0,-c) where c² = a² - b².