Conic Sections — Print
Slice a cone four ways and you get every curve in this unit.
What the practice questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Coefficient — The number multiplying a variable — the 3 in 3n.
- Open — Open excludes the endpoint (<, >); closed includes it (≤, ≥).
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Radius — Centre to edge. Half the diameter.
- Parabola — The U-shaped graph of a quadratic function.
- Vertex — A turning point of the curve, on its axis.
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.
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.
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.
- Both variables are squared and both coefficients are positive. Same sign, so closed.
- Divide by 36: x29 + y24 = 1. Standard form needs a 1 on the right.
- 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.
- Group: (x2-6x) + (y2+4y) = 3. Constants to the right.
- Complete both squares: (x-3)2 - 9 + (y+2)2 - 4 = 3. Half of -6 squared is 9; half of 4 squared is 4.
- 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?
- The y2 term is the positive one. The minus sits on the x term.
- The positive term names the direction the branches open. That is the rule for every hyperbola.
- 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.
Problem 1
What curve is x2+y2=25?
- A circle
- An ellipse
- A parabola
- A hyperbola
Worked answer
A circle
Radius 5, centred at the origin
Problem 2
What curve is y = x2 - 4?
- A parabola
- A circle
- An ellipse
- A hyperbola
Worked answer
A parabola
Only x is squared, so it is a parabola
Problem 3
What curve is x29 - y24 = 1?
- A hyperbola
- An ellipse
- A circle
- A parabola
Worked answer
A hyperbola
Opposite signs open the curve out
Problem 4
Give the radius of x2+y2=49.
Worked answer
7
√49 = 7
Problem 5
Give the centre of (x-2)2+(y+5)2=9 as an ordered pair.
Worked answer
(2,-5)
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Problem 6
For x225+y216=1, how far does the curve reach along the x-axis?
Worked answer
5
√25 = 5
Problem 7
For the same ellipse, how far does it reach along the y-axis?
Worked answer
4
√16 = 4
Problem 8
Which way does y216-x29=1 open?
- Up and down
- Left and right
- All four ways
- It does not open
Worked answer
Up and down
The positive term names the opening direction
Problem 9
Give the vertex of y=(x-3)2+5 as an ordered pair.
Worked answer
(3,5)
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Problem 10
Find the radius of x2+y2-6x+4y-3=0.
Worked answer
4
(x-3)2+(y+2)2 = 3+9+4 = 16
r = 4
Problem 11
Find the centre of x2+y2-6x+4y-3=0 as an ordered pair.
Worked answer
(3,-2)
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Problem 12
What curve is 4x2+9y2=36?
- An ellipse
- A circle
- A hyperbola
- A parabola
Worked answer
An ellipse
Dividing by 36 gives x29+y24=1
Problem 13
For 4x2+9y2=36, how far does the curve reach along the x-axis?
Worked answer
3
x29+y24=1
√9 = 3
Problem 14
Which way does x = 2y2 open?
- Right
- Up
- Left
- Down
Worked answer
Right
y is squared, so the axis is horizontal
The positive coefficient opens it right
Problem 15
A circle has centre (1,-4) and radius 6. Give the constant on the right of its standard equation.
Worked answer
36
62 = 36
Problem 16
Changing the minus to a plus in x29-y24=1 turns it into what?
- An ellipse
- A parabola
- A circle
- A line
Worked answer
An ellipse
Two positive squared terms with different denominators give an ellipse