Sections of a cone , the four conics

Take an infinite double cone , two cones joined at their tips, opening up and down. Slice it with a plane. Depending on the angle of the plane relative to the cone's axis, the cross-section is one of four shapes:

Plane perpendicular to the axis: a circle.
Plane tilted but cutting only one nappe: an ellipse.
Plane parallel to a generator (a line drawn on the cone's surface): a parabola.
Plane cutting both nappes: a hyperbola (two branches).

These four curves are collectively called conic sections.

The focus-directrix definition

Beyond the cone-slicing definition, every (non-circular) conic admits a beautiful unified description.

Definition. Given a point $F$ (the focus) and a line $\ell$ (the directrix) not passing through $F$ , a conic is the set of points $P$ such that $\frac{|PF|}{d(P, \ell)} = e,$ where $e \ge 0$ is a constant called the eccentricity.

$0 < e < 1$ : ellipse (closed curve, bounded).
$e = 1$ : parabola (open curve, one branch).
$e > 1$ : hyperbola (two branches, open).
$e = 0$ : circle (degenerate limiting case , focus is the centre, directrix at infinity).

The eccentricity measures how "non-circular" the conic is. A circle has $e = 0$ ; a very elongated ellipse has $e$ close to $1$ ; a parabola has $e = 1$ exactly; hyperbolas have $e > 1$ .

Standard placements

For tractable algebra, we place each conic so that its axis of symmetry lies along the $x$ -axis (or $y$ -axis) and its centre (or vertex) is at the origin.

Conic	Standard form	Eccentricity
Circle	$x^2 + y^2 = r^2$	$0$
Parabola	$y^2 = 4 a x$	$1$
Ellipse	$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ( $a > b$ )	$\sqrt{1 - b^2/a^2}$
Hyperbola	$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$	$\sqrt{1 + b^2/a^2}$

Why conics matter , the physics

Planetary orbits. Kepler's first law (1609): planets move in ellipses with the Sun at one focus.
Projectiles. Neglecting air resistance, any projectile travels along a parabola.
Satellite dishes & headlights. A paraboloid (3D parabola) reflects parallel rays through its focus.
Cooling towers. Many use a hyperboloid shape for structural strength.

So the four conics are everywhere in physics and engineering, not just on JEE papers.

Worked examples

Example 1. A point moves so that its distance from $(1, 0)$ equals its distance from the line $x = -1$ . What curve does it trace?

Distance from $(1, 0)$ : $\sqrt{(x - 1)^2 + y^2}$ . Distance from $x = -1$ : $|x + 1|$ . Set equal, square: $(x - 1)^2 + y^2 = (x + 1)^2 \Rightarrow y^2 = 4 x$ . A parabola.

Example 2. Find the eccentricity of $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ .

$a^2 = 25, b^2 = 9$ , $c^2 = a^2 - b^2 = 16, c = 4$ . So $e = c/a = 4/5$ . Ellipse.

Example 3. What kind of conic is $4 x^2 + 9 y^2 = 36$ ?

Divide by $36$ : $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ . Ellipse with $a^2 = 9, b^2 = 4$ .

Example 4. Identify $x^2 - y^2 = 16$ .

$\dfrac{x^2}{16} - \dfrac{y^2}{16} = 1$ . Hyperbola with $a = b = 4$ . Asymptotes $y = \pm x$ .

Example 5. A point is twice as far from $(2, 0)$ as from $x = -2$ . What conic?

$\sqrt{(x - 2)^2 + y^2} = 2 |x + 2|$ . So $e =$ (distance to focus)/(distance to directrix) , wait, here the ratio is reversed. Let's set it: distance to point / distance to line = $1/2 < 1$ ? No, the ratio is $2$ (distance to focus is twice that to line), so this is a hyperbola with $e = 2$ .

Square: $(x - 2)^2 + y^2 = 4(x + 2)^2 \Rightarrow x^2 - 4x + 4 + y^2 = 4 x^2 + 16 x + 16 \Rightarrow 3 x^2 + 20 x + 12 - y^2 = 0$ . This is a hyperbola.

Try it yourself

Identify the conic $x^2 + y^2 = 9$ .
Identify the conic $y^2 = 8 x$ .
Identify the conic $\dfrac{x^2}{16} + \dfrac{y^2}{25} = 1$ . (Note: $b > a$ , vertical major axis.)
Identify the conic $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$ .
Find eccentricity of $\dfrac{x^2}{36} + \dfrac{y^2}{25} = 1$ .
Find eccentricity of $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$ .
A point moves so that its distance from $(0, 3)$ equals its distance from $y = -3$ . Find the equation.
A point's distance from origin is half its distance from $x = 4$ . Find the conic.
Identify $9 x^2 - 16 y^2 = 144$ .
What is the eccentricity of a circle?
Convert $4 x^2 + 25 y^2 = 100$ to standard form and identify.
Show that for a parabola, eccentricity is always $1$ .

Pitfalls / Tricks

Always reduce a general second-degree equation to one of the standard forms before identifying.
Check whether $a^2 > b^2$ or $a^2 < b^2$ in an ellipse , the major axis is along the larger one.
Hyperbola requires the difference of squares to equal $1$ , with opposite signs.
Insight. Eccentricity is the single number that distinguishes the four conics. It is the deepest unifying parameter in this chapter.