TN 12th Maths — Learn, Visualize, Practice

1. What is a conic?

Slice a double cone with a plane and the curve of the cut is a conic section. Tilt the plane and you move smoothly between a circle, an ellipse, a parabola, and a hyperbola. Remarkably, all four are captured by a single focus–directrix definition.

DefinitionFocus–directrix definition

A conic is the set of points $P$ whose distance to a fixed point (the focus) and a fixed line (the directrix) are in a constant ratio $e$ , the eccentricity: $\dfrac{PF}{PM} = e$ , where $PM$ is the perpendicular distance to the directrix.

2. Eccentricity decides the shape

That single number $e$ determines which of the four conics you get. It is the most important quantity in the chapter.

$e = 0$ → circle
$0 < e < 1$ → ellipse
$e = 1$ → parabola
$e > 1$ → hyperbola

Key idea: In the Visualize tab, switch between the four conics and watch $e$ and the foci move — the same object, deformed.

3. Circle

The simplest conic: all points a fixed distance $r$ from a centre. There is no elongation, so $e = 0$ .

Centre at origin

x^2 + y^2 = r^2

Centre (h, k)

(x-h)^2 + (y-k)^2 = r^2

Example

Find the centre and radius of

x^2 + y^2 - 4x + 6y - 12 = 0

.

4. Parabola

Every point is equidistant from the focus and the directrix, so $e = 1$ . The standard right-opening parabola:

y^2 = 4ax

Focus $(a, 0)$ , directrix $x = -a$ , vertex $(0,0)$
Length of the latus rectum (the focal chord perpendicular to the axis) is $4a$

Example

For

y^2 = 12x

, find the focus, directrix and latus rectum.

5. Ellipse

An ellipse is the set of points whose distances to two foci add to a constant. It looks like a stretched circle with $0 < e < 1$ .

a > b, major axis along x

\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

$c = \sqrt{a^2 - b^2}$ , foci at $(\pm c, 0)$
Eccentricity $e = \dfrac{c}{a} < 1$
$a$ = semi-major axis, $b$ = semi-minor axis

Example

Find the foci and eccentricity of

\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1

.

6. Hyperbola

A hyperbola is the set of points whose distances to two foci differ by a constant. It has two branches and a pair of straight-line asymptotes it approaches at infinity; here $e > 1$ .

\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1

$c = \sqrt{a^2 + b^2}$ (note the plus sign), foci $(\pm c, 0)$
Eccentricity $e = \dfrac{c}{a} > 1$
Asymptotes $y = \pm\dfrac{b}{a}x$

Note: The only algebraic difference from the ellipse is the minus sign — and it changes $c = \sqrt{a^2 - b^2}$ into $c = \sqrt{a^2 + b^2}$, pushing the foci outside the curve.

7. Real-life conics

Parabola — projectile paths, satellite-dish and headlight reflectors (rays through the focus reflect parallel).
Ellipse — planetary orbits (Kepler: the Sun sits at one focus), whispering galleries.
Hyperbola — the shape of a cooling tower, and long-range navigation (LORAN) based on time differences.