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 $F$ ) and a fixed line (the directrix) are in a constant ratio $e$ , called the eccentricity: $\dfrac{PF}{PM} = e$ , where $PM$ is the perpendicular distance from $P$ to the directrix.

A single second-degree equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ describes every conic. At this level we work with $B = 0$ (no $xy$ term), so the axes stay parallel to the coordinate axes.

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 — most 'identify the conic' questions come straight from it.

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

You can read the type straight off the general equation without completing the square:

$A = C$ (and no $xy$ term) → circle.
$B = 0$ and exactly one of $A, C$ is zero → parabola.
$A \ne C$ but $A, C$ have the same sign → ellipse.
$A, C$ have opposite signs → hyperbola.

Example

Identify the conic: (a)

16y^2 = -4x^2 + 64

, (b)

x^2 - y^2 = x + 3

, (c)

4x^2 - 9y^2 - 16x + 18y = 29

.

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

3. Circle — standard and general form

A circle is the locus of a point that stays a fixed distance $r$ from a fixed 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

Expanding the centre–radius form gives the general form. Compare it carefully — this is the workhorse identity of the whole circle section:

General form

x^2 + y^2 + 2gx + 2fy + c = 0

Centre $(-g, -f)$
Radius $r = \sqrt{g^2 + f^2 - c}$
Real circle if $g^2 + f^2 - c > 0$ ; a single point if $=0$ ; no real locus if $<0$

Note: An equation is a circle only when the coefficients of $x^2$ and $y^2$ are equal and there is no $xy$ term. If they differ, divide through first so both become $1$ before you read off $g$ and $f$.

Example

Find the general equation of the circle with centre

(-3, -4)

and radius

3

.

ExampleCompleting the square

Find the centre and radius of

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

.

4. Circle — diameter form and position of a point

If you know the two endpoints of a diameter, you never need to find the centre and radius separately — this compact form gives the circle at once.

DefinitionDiameter form

If $(x_1, y_1)$ and $(x_2, y_2)$ are the ends of a diameter, the circle is $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$ .

Example

Find the circle whose diameter joins

(-4, -2)

and

(1, 1)

.

ExampleLine meets the axes

The line

3x + 4y - 12 = 0

meets the axes at

A

and

B

. Find the circle on

AB

as diameter.

DefinitionPosition of a point (Theorem 5.3)

Substitute $P(x_1, y_1)$ into the general form. Then $P$ is outside, on, or inside the circle according as $x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$ is $> 0$ , $= 0$ , or $< 0$ .

Example

Where does

(2, 3)

lie relative to

x^2 + y^2 - 6x - 8y + 12 = 0

?

5. Parabola — the four standard orientations

Every point of a parabola is equidistant from the focus and the directrix, so $e = 1$ . The vertex sits halfway between them. There are four standard parabolas with vertex at the origin, one for each direction of opening.

Vertex at the origin, focal distance a > 0

$y^2 = 4ax$ — opens right; focus $(a,0)$ , directrix $x = -a$ , axis the $x$ -axis.
$y^2 = -4ax$ — opens left; focus $(-a,0)$ , directrix $x = a$ .
$x^2 = 4ay$ — opens up; focus $(0,a)$ , directrix $y = -a$ , axis the $y$ -axis.
$x^2 = -4ay$ — opens down; focus $(0,-a)$ , directrix $y = a$ .

DefinitionLatus rectum

The latus rectum is the focal chord perpendicular to the axis. For every standard parabola its length is $4a$ — and its endpoints are exactly $2a$ above and below the focus. It fixes how 'wide' the parabola is at the focus.

Example

For

y^2 = 12x

, find the vertex, focus, directrix and latus rectum.

ExampleBuilding the equation from focus & directrix

Find the parabola with focus

(-2, 0)

and directrix

x = 2

.

ExampleShifted vertex

Find the vertex, focus, directrix and latus rectum of

x^2 - 4x - 5y - 1 = 0

.

6. Ellipse — foci, vertices, directrices, latus rectum

An ellipse is the set of points whose distances to two foci add to a constant $2a$ . 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

$c = \sqrt{a^2 - b^2}$ (minus sign), foci $(\pm c, 0)$
Eccentricity $e = \dfrac{c}{a} < 1$
Vertices $(\pm a, 0)$ ; ends of minor axis $(0, \pm b)$
Directrices $x = \pm\dfrac{a}{e}$
Length of latus rectum $= \dfrac{2b^2}{a}$

Note: If the larger denominator is under $y^2$ instead, the major axis is vertical: swap the roles, so foci become $(0, \pm c)$ and directrices $y = \pm\tfrac{a}{e}$. Always let $a$ be the larger semi-axis.

Example

Find the foci and eccentricity of

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

.

ExampleFrom foci and vertices

Find the ellipse with foci

(\pm 2, 0)

and vertices

(\pm 3, 0)

.

ExampleCompleting the square

For

4x^2 + y^2 + 24x - 2y + 21 = 0

, find the centre, vertices and foci.

7. Hyperbola — foci, asymptotes, directrices

A hyperbola is the set of points whose distances to two foci differ by a constant $2a$ . It has two branches and a pair of straight-line asymptotes it hugs at infinity; here $e > 1$ .

Transverse axis along x

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

$c = \sqrt{a^2 + b^2}$ (plus sign), foci $(\pm c, 0)$
Eccentricity $e = \dfrac{c}{a} > 1$ , equivalently $e = \sqrt{1 + \tfrac{b^2}{a^2}}$
Vertices $(\pm a, 0)$ ; transverse axis length $2a$ , conjugate axis $2b$
Asymptotes $y = \pm\dfrac{b}{a}x$
Directrices $x = \pm\dfrac{a}{e}$ ; latus rectum $= \dfrac{2b^2}{a}$

Note: The only algebraic difference from the ellipse is the minus sign — and it flips $c = \sqrt{a^2 - b^2}$ into $c = \sqrt{a^2 + b^2}$, pushing the foci outside the curve. Note also that here $a$ is the number under the positive term, whether or not it is the larger.

Example

Find the vertices and foci of

9x^2 - 16y^2 = 144

.

ExampleTransverse axis on the y-axis

Find the hyperbola with vertices

(0, \pm 4)

and foci

(0, \pm 6)

.

ExampleCompleting the square

Find the centre, foci and eccentricity of

11x^2 - 25y^2 - 44x - 50y - 256 = 0

.

8. Parametric forms

Instead of a relation between $x$ and $y$ , a single parameter can generate every point of a conic. Parametric points make tangent and locus problems much cleaner — one variable instead of two.

Circle $x^2 + y^2 = a^2$ : $\ (a\cos\theta,\ a\sin\theta)$
Parabola $y^2 = 4ax$ : $\ (at^2,\ 2at)$
Ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ : $\ (a\cos\theta,\ b\sin\theta)$
Hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ : $\ (a\sec\theta,\ b\tan\theta)$

Note: For the ellipse, $\theta$ (the eccentric angle) is the angle at the centre to the corresponding point on the auxiliary circle $x^2+y^2=a^2$ — not the angle to the point on the ellipse itself.

Example

Verify that

(a\cos\theta, b\sin\theta)

lies on

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

.

9. Tangents and normals to conics

A tangent touches the curve at exactly one point; the normal is perpendicular to it through the same point. For the parabola $y^2 = 4ax$ the tangent at $(x_1, y_1)$ has a neat replacement rule (replace $y^2$ by $yy_1$ and $2x$ by $x + x_1$ ):

Parabola tangent at (x₁, y₁)

yy_1 = 2a(x + x_1)

Parabola y² = 4ax, point (at², 2at)

Tangent (parametric): $ty = x + at^2$
Normal (parametric): $y + tx = 2at + at^3$

For the ellipse and hyperbola the slope form is the exam favourite — memorise the sign difference (the only change is $+b^2$ vs $-b^2$ ):

Ellipse tangent of slope m

y = mx \pm \sqrt{a^2 m^2 + b^2}

Hyperbola tangent of slope m

y = mx \pm \sqrt{a^2 m^2 - b^2}

ExampleTangent & normal to a parabola

Find the tangent and normal to

x^2 + 6x + 4y + 5 = 0

at

(1, -3)

.

10. Condition of tangency

When does the line $y = mx + c$ just touch a conic? Substituting and forcing the resulting quadratic to have equal roots gives a single condition linking $c$ to the slope $m$ . These are heavily tested.

Parabola $y^2 = 4ax$ : tangent is $y = mx + \dfrac{a}{m}$ , i.e. $c = \dfrac{a}{m}$ ; point of contact $\left(\dfrac{a}{m^2}, \dfrac{2a}{m}\right)$ .
Ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ : condition $c^2 = a^2 m^2 + b^2$ .
Hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ : condition $c^2 = a^2 m^2 - b^2$ .

Example

For what

c

is

y = 2x + c

a tangent to

\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1

?

Watch out: Two tangents can be drawn to any conic from an external point. The locus of points from which the two tangents are perpendicular is the directrix for a parabola, and the director circle $x^2 + y^2 = a^2 \pm b^2$ for an ellipse ($+$) or hyperbola ($-$).

11. Real-life conics

The reflective properties of conics power much of the technology around you. A ray from the focus of a parabola reflects out parallel to the axis (and vice-versa); a ray from one focus of an ellipse or hyperbola reflects to the other focus.

Parabola — headlight and satellite-dish reflectors, solar heating troughs, suspension-bridge cables, and projectile paths.
Ellipse — planetary orbits (Kepler: the Sun sits at one focus), whispering galleries, and the lithotripter that shatters kidney stones with waves focused from one focus to the other.
Hyperbola — cooling-tower profiles, long-range LORAN navigation from time differences, and the paths of some comets.

ExampleSuspension-bridge / parabolic arch

A parabolic arch is

15\text{ m}

high at the centre and

40\text{ m}

wide at the base. Find its equation with the vertex at the origin.

ExampleOrbit — distance between foci

Earth's maximum and minimum distances from the Sun are

152 \times 10^6

km and

94.5 \times 10^6

km. The Sun is at one focus. Find the distance between the two foci.

ExampleWhispering gallery (ellipse)

An elliptical ceiling is

34\text{ m}

long and

8\text{ m}

high at the centre. Where are the foci?