Conic Sections
Circle
The equation of a circle with radius r is
y2
x2
+
=1
r2
r2
(x − h)2
(y − k)2
+
=1
2
r
r2
(a) Circle centered at (0, 0)
(centered at the origin (0, 0))
(centered at the point (h, k))
(b) Circle centered at (h, k)
Parabola
The equation of a parabola with axis parallel to the x-axis is
y 2 = 4px
(vertex at the origin (0, 0))
2
(y − k) = 4p(x − h)
(vertex at the point (h, k).)
The focus is the point
(p, 0)
(vertex at the origin (0, 0))
(h + p, k)
(vertex at the point (h, k).)
The directrix is the line
x=−p
(vertex at the origin (0, 0))
x =h − p
(a) Parabola with vertex at (0, 0) (p positive)
(vertex at the point (h, k).)
(b) Parabola with vertex at (h, k) (p negative)
The equation of a parabola with axis parallel to the y-axis is
x2 = 4py
(vertex at the origin (0, 0))
2
(x − h) = 4p(y − k)
(vertex at the point (h, k)).
The focus is the point
(0, p)
(vertex at the origin (0, 0))
(h, k + p)
(vertex at the point (h, k)).
The directrix is the line
y =−p
(vertex at the origin (0, 0))
y =k − p
(a) Parabola with vertex at (0, 0) (p negative)
(vertex at the point (h, k)).
(b) Parabola with vertex at (h, k) (p positive)
Ellipse
The equation of an ellipse with major axis parallel to the x-axis is
y2
x2
+
=1
a2
b2
2
2
(x − h)
(y − k)
+
=1
2
a
b2
(centered at the origin (0, 0))
(centered at the point (h, k))
where a ≥ b > 0.
The vertices are at
(±a, 0) and (0, ±b)
(centered at the origin (0, 0))
(h ± a, k) and (h, k ± b)
(centered at the point (h, k)).
The foci are at
(±c, 0)
(centered at the origin (0, 0))
(h ± c, k)
(centered at the point (h, k))
where c2 = a2 − b2 .
Figure: Ellipse centered at (0, 0) with major axis parallel to the x-axis.
The equation of an ellipse with major axis parallel to the y-axis is
y2
x2
+ 2 = 1 (centered at the origin (0, 0))
2
a
b
(y − k)2
(x − h)2
+
= 1 (centered at the point (h, k))
a2
b2
where a ≥ b > 0.
The vertices are at
(±b, 0) and (0, ±a)
(centered at the origin (0, 0))
(h ± b, k) and (h, k ± a)
(centered at the point (h, k)).
The foci are at
(0, ±c)
(centered at the origin (0, 0))
(h, k ± c)
(centered at the point (h, k))
where c2 = a2 − b2 .
Figure: Ellipse centered at (0, 0) with major axis parallel to the y-axis.
Hyperbola
The equation of a hyperbola with axis parallel to the x-axis is
y2
x2
−
=1
a2
b2
2
2
(x − h)
(y − k)
−
=1
2
a
b2
(centered at the origin (0, 0))
(centered at the point (h, k)).
The vertices are at
(±a, 0)
(centered at the origin (0, 0))
(h ± a, k)
(centered at the point (h, k)).
(±c, 0)
(centered at the origin (0, 0))
(h ± c, k)
(centered at the point (h, k))
The foci are at
where c2 = a2 + b2 .
The asymptotes are at
b
y = ± x (centered at the origin (0, 0))
a
b
y = ± (x − h) + k (centered at the point (h, k)).
a
Figure: Hyperbola centered at (0, 0) with axis parallel to the x-axis.
The equation of a hyperbola with axis parallel to the y-axis is
y2
x2
− 2 =1
2
a
b
(y − k)2
(x − h)2
−
=1
a2
b2
(centered at the origin (0, 0))
(centered at the point (h, k)).
The vertices are at
(0, ±a)
(centered at the origin (0, 0))
(h, k ± a)
(centered at the point (h, k)).
(0, ±c)
(centered at the origin (0, 0))
(h, k ± c)
(centered at the point (h, k))
The foci are at
where c2 = a2 + b2 .
The asymptotes are at
a
y = ± x (centered at the origin (0, 0))
b
a
y = ± (x − h) + k (centered at the point (h, k))
b
Figure: Hyperbola centered at (0, 0) with axis parallel to the y-axis.