Notes for Geometry Conic Sections

The notes is taken from Geometry, by David A. Brannan, Matthew F. Esplen and Jeremy J. Gray, 2nd edition

1 Conic Sections

A conic section is deﬁned as the curve of intersection of a double cone with a plane. Figure:

Examples:

1. non-degenerate conic sections: parabolas, ellipses or hyperbolas;

2. degenerate conic sections: the single point, single line and pair of lines.

1.1 Focus-Directrix Definition of the Non-Degenerate Conics The 3 non-degenerate conics can be defined as the set of points P in the plane satisfying: The distance of P from a fixed point F (called the focus of the conic) is a constant multiple (called its eccentricity, e) of the distance of P from a fixed line d (called its directrix).

Eccentricity: A non-degenerate conic is

1. an ellipse if 0 ≤ e < 1,

2. a parabola if e = 1,

3. a hyperbola if e > 1.

1 Parabola (e = 1): A parabola is defined to be the set of points P in the plane whose distance from a fixed point F is equal to their distance from a fixed line d.

We now derive a parabola in standard form:

Let F (a, 0) be the focus and d : x = −a be the directrix. Let P (x, y) be an arbitrary point on the parabola and let M(−a, y) be the foot of the perpendicular from P to the directrix. Figure:

Since FP = PM, by the deﬁnition of the parabola, this follows that

FP 2 = PM 2 ⇒ (x − a)2 + y2 = (x + a)2 ⇒ y2 = 4ax.

The point (at2, 2at), t ∈ R lies on the parabola.

2 2 ∵ (2at) = 4a · at . Conversely, we can write the coordinates of each point on the parabola in the form 2 y y2 (2at)2 2 (at , 2at). For if we choose t = 2a , then y = 2at and x = 4a = 4a = at . It follows that there is a one-to-one correspondence between the real numbers t and the points of the parabola. Parabola in standard form A parabola in standard form has equation

y2 = 4ax, where a > 0.

It has focus (a, 0) and directrix x = −a; and it can be described by the parametric equations: 2 x = at , y = 2at (t ∈ R). We call the x-axis the axis of the parabola in standard form, since the parabola is symmetric with respect to this line. We call the origin the vertex of a parabola in standard form, since it is the point of the intersection of the axis of the parabola with the parabola. A parabola has no center.

2 Example 1.1. Write down the focus, vertex, axis and directrix of the parabola E with equation y2 = 2x.

1 1 Solution. Focus: F = ( 2 , 0), Axis: x-axis, Vertex: (0, 0), Directrix: x = − 2 .

Ellipse (0 ≤ e < 1): We define an ellipse with eccentricity zero to be a circle. We define an ellipse with eccentricity e (where 0 < e < 1) to be the set of points P in the plane whose distance from a fixed point F is e times their distance from a fixed line.

We now derive an ellipse in standard form:

a Let F (ae, 0), a > 0 be the focus and d : x = e . Let P (x, y) be an arbitrary point on a the parabola and let M( e , y) be the foot of the perpendicular from P to the directrix. Figure:

Since FP = e · PM, by the deﬁnition of the ellipse, this follows that

a x2 y2 FP 2 = e2 · PM 2 ⇒ (x − ae)2 + y2 = e2(x − )2 ⇒ + = 1. e a2 a2(1 − e2) √ Let b = a 1 − e2. Then x2 y2 + = 1. a2 b2 The equation is symmetrical in x and y. The ellipse also has a second focus F 0(−ae, 0) 0 a and a second directrix d : x = − e . We call the segment joining the points (±a, 0) the major axis of the ellipse and the segment joining the points (0, ±b) the minor axis of the ellipse. ∵ b < a, the minor axis is shorter than the major axis. The origin is the center of this ellipse. Note that (a cos t, b sin t) lies on the ellipse.

(a cos t)2 (b sin t)2 + = cos2 t + sin2 t = 1. ∵ a2 b2

3 We can check that x = a cos t, y = b sin t, t ∈ (−π, π] gives a parametric representation of the ellipse. Ellipse in standard form An ellipse in standard form has equation x2 y2 + = 1, where a ≥ b > 0, b2 = a2(1 − e2), 0 ≤ e < 1. a2 b2 It can be described by the parametric equations

x = a cos t, y = −b sin t, t ∈ (−π, π].

a If e > 0, it has foci (±ae, 0) and directrix x = ± e .

Exercise 1.2. p(x + ae)2 + y2 + p(x − ae)2 + y2 = 2a.

Hyperbola (e > 1): A hyperbola is the set of points P in the plane whose distance from a ﬁxed point F is e times their distance from a ﬁxed line d, where e > 1.

We now derive a hyperbola in standard form:

a Let F (ae, 0), a > 0 be the focus and d : x = e . Let P (x, y) be an arbitrary point on a the parabola and let M( e , y) be the foot of the perpendicular from P to the directrix. Figure:

Since FP = e · PM, by the deﬁnition of the hyperbola, this follows that a x2 y2 FP 2 = e2 · PM 2 ⇒ (x − ae)2 + y2 = e2(x − )2 ⇒ − = 1. e a2 a2(e2 − 1) √ Let b = a e2 − 1. Then x2 y2 − = 1. a2 b2

4 The equation is symmetrical in x and y. The ellipse also has a second focus F 0(−ae, 0) 0 a and a second directrix d : x = − e . We call the segment joining the points (±a, 0) the major (transverse) axis of the hyperbola and the segment joining the points (0, ±b) the minor (conjugate) axis of the hyperbola. The origin is the center of this hyperbola. π π π 3π We can check that x = sec t, y = b tan t, t ∈ (− 2 , 2 ) ∪ ( 2 , 2 ) gives a parametric representation of the hyperbola. The hyperbola consist of 2 branches. When x → b ±∞, the branches get closer and closer to the lines y = ± a x the asymptotes of the hyperbola. Hyperbola in standard form A hyperbola in standard form has equation

x2 y2 − = 1, where a ≥ b > 0, b2 = a2(e2 − 1), e > 1. a2 b2

a it has foci (±ae, 0) and directrix x = ± e . It can be described by the parametric equations π π π 3π x = a sec t, y = −b tan t, t ∈ (− , ) ∪ ( , ). 2 2 2 2

Example 1.3. Determine the foci F and F 0 of the hyperbola E with equation x2 − 2y2 = 1.

Solution. ±p3/2, 0 .

Exercise 1.4. p(x + ae)2 + y2 − p(x − ae)2 + y2 = ±2a.