Conic Sections

Slice a cone four ways and you get every curve in this unit.

Take a cone and cut straight through it, parallel to the base.

The edge of the cut is a circle. Nobody needed an equation to know that.

one cone, four different cuts circleellipseparabola tilt the knife and the edge changes shape cut steeper still and the curve never closes

Now tilt the knife slightly. The circle stretches into an ellipse — still closed, no longer round.

Tilt further, until the cut runs parallel to the side of the cone, and the curve never closes at all.

Circle, ellipse, parabola, hyperbola — one solid, one knife, four curves. They share a family resemblance in their equations because they share an origin.

The four conic sections and the equation each one has circle x² + y² = r² ellipse x²/a² + y²/b² = 1 parabola y = ax² + bx + c hyperbola x²/a² − y²/b² = 1 tell them apart by the signs: same sign is closed, opposite signs open out

Telling them apart from the equation

Look at the squared terms. One squared term means a parabola — only x or only y is squared.

Two squared terms with the same sign give a circle or an ellipse; equal coefficients make it a circle.

Two squared terms with opposite signs give a hyperbola, which opens out and never closes.

Three questions, four answers

How many squares? Same sign or opposite? Equal coefficients? For axis-aligned, nondegenerate conics with no xy term, those three answers identify the type. Degenerate equations can instead describe a point, lines or no real points before a graph is drawn.

Circles and ellipses

(x-h)2a2 + (y-k)2b2 = 1 is an ellipse centred at (h,k).

It reaches a either side horizontally and b vertically, so the bigger denominator names the long direction.

When a = b the two reaches are equal and the ellipse is a circle.

The denominators are squares

In x225 the reach is 5, not 25. Take the square root before drawing.

Parabolas

y = a(x-h)2 + k opens up or down; x = a(y-k)2 + h opens left or right.

For y=a(x-h)2+k, the axis of symmetry is x=h; for x=a(y-k)2+h, it is y=k. Both coordinates can be unbounded.

Every point on the curve is equally far from the focus and from a fixed line called the directrix.

Which way it opens

If y is alone, the parabola opens vertically; if x is alone, horizontally. The sign of a then picks which of the two.

Hyperbolas

(x-h)2a2 - (y-k)2b2 = 1 opens left and right; swapping which term is positive opens it up and down.

The positive term names the opening direction, which is the only thing you need to sketch it.

Far from the centre the two branches settle onto straight asymptotes, which is why they never close.

The minus decides everything

Changing - to + turns a hyperbola into an ellipse — a curve that opens forever into one that closes. Check the sign before anything else.

Completing the square to find the centre

Equations often arrive expanded, like x2 + y2 - 6x + 4y - 3 = 0, with the centre hidden.

Group the x terms and the y terms, complete the square on each, and the centre appears in the brackets.

It is the same move as in Algebra II, done twice — once for each variable.

Balance both sides

Whatever you add inside to complete a square must be added to the right too. Forgetting that changes the radius while leaving the centre correct, which makes it hard to spot.

Worked examples

Identifying

Identify the curve 4x2 + 9y2 = 36.

  1. Both variables are squared and both coefficients are positive. Same sign, so closed.
  2. Divide by 36: x29 + y24 = 1. Standard form needs a 1 on the right.
  3. The coefficients differ, so it is an ellipse reaching 3 across and 2 up. Square roots of the denominators.

An ellipse, a=3, b=2

Finding the centre

Find the centre and radius of x2 + y2 - 6x + 4y - 3 = 0.

  1. Group: (x2-6x) + (y2+4y) = 3. Constants to the right.
  2. Complete both squares: (x-3)2 - 9 + (y+2)2 - 4 = 3. Half of -6 squared is 9; half of 4 squared is 4.
  3. So (x-3)2+(y+2)2 = 16: centre (3,-2), radius 4. Radius is the square root of 16.

Centre (3,-2), radius 4

Reading a hyperbola

Which way does y216 - x29 = 1 open?

  1. The y2 term is the positive one. The minus sits on the x term.
  2. The positive term names the direction the branches open. That is the rule for every hyperbola.
  3. So it opens up and down, with vertices 4 above and below the centre. √16=4.

Up and down

Words to know

Conic section
A curve obtained by slicing a cone with a plane.
Focus
A fixed point used to define the curve. An ellipse and a hyperbola have two.
Vertex
A turning point of the curve, on its axis.
Major axis
The longer axis of an ellipse.
Eccentricity
How far a conic is from being a circle. Zero is a circle; one is a parabola.

Mistakes to watch for

Reading the denominator as the reach. In x225=1 the curve reaches 5, not 25 — the denominators are squares and need a root taken before sketching.

Mistaking a hyperbola for an ellipse. The sign between the two squared terms is the whole difference: a plus closes the curve, a minus sends it off to infinity.

Deciding the opening direction of a hyperbola from the larger denominator. It is the positive term that names the direction, whatever the numbers underneath happen to be.

Forgetting to balance while completing the square. Adding 9 inside the bracket means adding 9 to the other side too; otherwise the centre is right and the radius is wrong.

Assuming two squared terms always make a circle. Only equal coefficients give a circle; 4x2+9y2=36 is an ellipse, wider than it is tall.

Apollonius worked all this out around 200 BC, eighteen centuries before anyone had a use for it.

Kepler found that planets travel in ellipses with the sun at one focus. Newton then proved the ellipse follows from an inverse-square force.

A satellite dish is a parabola because every ray arriving parallel to its axis bounces to one point — the focus, where the receiver sits.

Browse all courses