Topic 4: ConicsWebsite: Click here or use

Achievement Standard 91573: Apply the geometry of conic sections in solving problems.

This standard is an Internal worth 3 credits.

Introduction to Conic sections. To be completed by Monday 23 July

Learning Objectives

  1. To identify applications for each of the 4 conic sections and describe these.
  2. To understand the four conic sections as the intersection of a cone and a plane.
  3. To understand the four conic sections as a locus of points.
  4. To review knowledge of the translations of a curve and how it changes the equation.

For the circle:

  1. Use the locus definition to derive the standard Cartesian equation and be able to adjust the equation for translations of the standard graph.
  2. Be able to use completing the square to find the equation in factored form if given in expanded form.
  3. Understand the standard equation in parametric form and be able to adjust it for translations of the standard graph.
  4. Be able to identify the key features of the curves from the equation or a graph.

Topic / Notes / Delta Maths 2nd Edition Old Book / Delta Maths NCEA Level 3 New Book
Transformations of curves / We covered transformations of curves in relation to trigonometric graphs. You need to review this work, in particular the horizontal and vertical translations. Note that a vertical translation can be written in 2 ways.

Both of these represent a vertical translation of positive a. / Page 21-24 / Page 500-501
Circles
/ A circle is the intersection of a cone and a horizontal plane.
A circle is the locus of a point that moves a fixed distance, in a plane, from a fixed point.
Using this locus definition and Pythagoras’ Theorem we can derive the equation for a standard circle. This can then easily be translated using our knowledge of the effect of translations on the equation of a graph.
This link allows you to see the effect of changes to the equation on the graph.
/ Read the next page for notes on the circle.
Read Page 357-359
Do…
Page 358
Ex 37.1 No 1, 2, 3, 5, 8 and 9.
Ex 37.2 No 1, 2 and 4
Page 379-380 Ex 38.3 / Read the next page for notes on the circle.
Read Page 7-10
Do…
Page 8
Ex 1.02 No 1, 2, 3, 5, 7, 10 and 11
Ex 1.03 No 1, 2 and 4
Page 40 Ex 2.03
Useful resources / 3
4
5
6The old Delta textbook Chapters 37 and 38. or the new Delta textbook Chapters 1 and 2

The Circle: (the highlighted page references are for the new Delta textbook)

  1. A circle is the intersection of a cone and a horizontal plane. (See page 355, page 5)
  1. A circle is the locus of a point that moves a fixed distance, in a plane, from a fixed point. (See Page 357, page 6-7)
  1. The standard equation is found by applying Pythagoras’ Theorem. (See page 357, page 6-7)

Features: Centre (0,0) and radius = r

The translated equation gives the circle for any other centre. (See page 357, page 6-7)

Features: Centre (a,b) and radius = r

This equation can be expanded to be of the form

We use our knowledge of perfect squares to put this back in factorised form so we can easily identify the centre and radius. (See page 357, page 6-8)

  1. The parametric form for a circle is found by applying trigonometry. (See page 379, page 39)

Features: Centre (0,0) and radius = r

This can be translated to any other centre.

Or

Features: Centre (a,b) and radius = r