**LESSON 26: PARABOLAS AND HYPERBOLAS**

**Do now:** Using the whole width of the page, draw axes with scales running from −10 to +10 for x and from −10 to +10 for y, and plot these curves

y^{2} = x

y^{2} = 4x

y = 16/x

y = 9/x

These are Parabolas and Hyperbolas. These are examples of Conic Sections, but in FP1 we look only at parabolas and hyperbolas, and only parabolas and hyperbolas arranged in a particular way like the ones we’ve sketched:

*Parabolas: axis along x-axis; vertex at origin*

*Hyperbolas: rectangular; asymptotes on y and x axes*

**Using x-t and y-t equations instead of x-y equations to describe curves**

Draw axes (using the whole width of the page) with x from −1 to +10, and y from −10 to +10. Plot the following graph:

At each of the times t=−6, t=−4, t=−2, t=0, t=2, t=4, t=6

x=¼t^{2}

y=½t

Which of the curves you’ve already drawn is this?

And this graph:

At each of the times t=−3, t=−2, t=−1, t=0, t=1, t=2, t=3

x=3t

y=^{3}⁄_{t}

Which of the curves you’ve already drawn is this?

What could be similar x-t and y-t equations for the other two curves you’ve already drawn?

Why do we bother with the extra variable t? Here’s an example. This curve can be neatly described by equations telling us the x and the y for each point in time t. It is more complicated to describe it by an x-y equation.

http://mathworld.wolfram.com/Cycloid.html

In this chapter you do not study the cycloid. You study other curves described by equations telling you the x and the y for each point in time t.

The “parametric forms” (x-t and y-t equations) we’ll use:

For the parabola y^{2}=4ax …….. x=at^{2}, y=2at

For the hyperbola xy=c^{2} …….. x=ct, y=^{c}⁄_{t}

**Classwork and homework**

Ex.3A Q.1, 3, 4 a and c, 5 a and c.

**Focus and directrix**

Worked solution: pdf. Also shown in image below.

The ancient Greeks worked out almost everything we know about conic sections without knowing any algebra. They defined a parabola as the path or **locus** of a point which was always the same distance from a particular point (called the **focus**) as it was from a particular line (called the **directrix**).

Our standard parabola described by

y^{2}=4ax, or

x=at^{2}, y=2at

has focus at x=a and directrix x=−a. We can prove this, and it is in the formula book.

All you have to know about focus and directrix for now is that you can find them in the formula book, and every point on a parabola is the same distance from the focus as from the directrix.

**Classwork and homework**

Ex.3B Q.1 a and b, 2 a and b, and 3.

**Lines crossing parabolas**

Example: Ex.3C Q.2.

First draw a diagram!

A bit better than just working with y and x is to say that the general point on the parabola is (8t^{2}, 16t).

Then the t-values for A and B will be the two solutions of the equation

16t = 8t^{2} + 6

or

4t^{2} − 8t + 3 = 0

(2t−3)(2t−1)=0, so t=1½ or ½

So A and B are (18,24) and (2,8)

Midpoint is (10,16)

Length AB (which the question doesn’t ask) is 16√2

**Classwork and homework**

Ex.3C Q.3 and 4

Review how to find the equation of a line connecting two points

Ex.3C Q.5, 6, 7

**Working with tangents and normals**

**Do now:** Draw tangent and normal to parabola described by (2t^{2}, 4t) at point where t=0.

Draw tangent and normal to hyperbola described by (t, 1/t) at point when t=1.

What is the slope of the tangent at a general point? dy/dx

What is the slope of the normal? −1/(slope of tangent)

dy/dx=(dy/dt)/(dx/dt) makes these problems neater. Why is it true?

Example: Ex.3D Q.1a

Draw a diagram

General point on parabola is (t^{2},2t)

dx/dt = 2t, dy/dt = 2, so dy/dx = 1/t (it’s always 1/t, for every parabola)

At the given point t=4, so slope of tangent = dy/dx = ¼

so tangent is y−8 = ¼ (x−16)

or x−4y+16=0 (check this looks right)

Example: Ex.3D Q.1c

Draw a diagram

General point on hyperbola is (5t, 5/t)

dy/dx = [−5/t^{2}]/5 = −1/t^{2} (it’s always −1/t^{2}, for every hyperbola)

At the given point, t=1, so slope of tangent = dy/dx = −1

so tangent is y−5 = −(x−5)

or x+y−10=0 (check this looks right)

**Classwork and homework**

Ex.3D Q.1 b, d, e, f and Q.2 and 3

The value of using dy/dx=(dy/dt)/(dx/dt) comes out in Ex.3E

Example: Ex.3E Q.1 and 2 using a instead of 3 and c instead of 6. Q.3 and Example 12 page 57 using these results.

Note: area.

If a tangent to the hyperbola at the point timestamped t meets the x-axis at X and the y-axis at Y, the area OXY is the same whatever t is.

Area of triangle = ½base×height. This is really three formulas, since every triangle has three bases and three heights.

**Classwork and homework**

Ex.3E Q.4, 5, 6, 7

**Trickier algebra**

Most exam questions are to do with meeting points. You need to remember these factorisations.

p^{2}−q^{2} = (p−q)(p+q) p^{2}+2pq+q^{2} = (p+q)^{2}

p^{2}−2pq+q^{2} = (p−q)^{2} also (for complex numbers) p^{2}+q^{2} = (p−qi)(p+qi)

p^{3}−q^{3} = (p−q)(p^{2}+pq+q^{2})

and you need to keep calm when solving equations where the coefficients are algebraic symbols.

Scaffolding for parabola-hyperbola questions

FIRST STEP: Draw a diagram. Put in the information from the question.

SECOND STEP is often to translate between the Cartesian equation of the curve (the x,y form) and the parametric form (the t-form). Use the formula book to help you with this. Use the formula book if you want to find the focus and directrix of a parabola. Every point on the parabola is the same distance from the focus point as from the directrix line.

THIRD STEP is usually to find a tangent (or a normal, or two tangents, or two normals…)

Find slope of tangent = dy/dx = (dy/dt) / (dx/dt).

Slope of normal is −1/slope of tangent

Then plug the slope and the point (at2,2at) or (ct, c/t) into y-y_{1}=m(x-x_{1}) or into y=mx+K. Or you may just be asked to find the equation of a line connecting two points on the curve (a chord).

FOURTH STEP may be to find where two tangents, or a tangent and a line, or two normals, or whatever, intersect. Use simultaneous equations for that.

Or it may be to find the points at which two tangents (or two normals) are drawn, given their point of intersection. For that, plug the x,y numbers for the point of intersection into your general equation for tangents (or normals), to turn that equation into an equation where t is now the unknown, not a constant. Solve to find two values of t.

If you’re asked where the tangent or a normal at the point where t=p meets the curve again, plug x=at^{2}, y=2at, or x=ct, y=c/t, into the tangent or normal equation and you will get a quadratic in t. You know t=p is one root of that quadratic. Use sum of roots to find the other root.

Look at context when thinking whether a letter represents a constant or a variable. If x=ct and y=c/t, then c is a constant (telling us which curve, and there’s only ever one curve in these FP1 questions) and t is a variable (telling time as we move along the curve).

When you get an equation for a tangent or normal at a point “timestamped” p, then along that individual tangent (or normal) p is a constant.

Question sheet about parametrics.

Homework questions on parametrics

Worked solutions here:

Harder parabola-hyperbola questions from 2014-2016 IAL papers, “scaffolded”, with answers.