Porisma

Conic sections

By the end, you can

  • Identify parabolas, ellipses and hyperbolas
  • Find centres, intercepts and asymptotes
  • Preserve restrictions when interpreting a conic as a function

about 20 minutes for the lesson · practise at your own pace.

The central idea

Conic equations describe curves that may fail the vertical-line test. In standard form, a sum of positive squared terms gives an ellipse; a difference gives a hyperbola. Completing the square identifies a shifted centre. A parabola has one squared coordinate and one linear coordinate.

A plane cuts a double cone to give four familiar conics.

Parallel to the circular base, it gives a circle. A tilted plane cutting just one nappe (half) without becoming parallel to a generator gives an ellipse.

A plane parallel to a generator gives a parabola.

A plane perpendicular to the base and cutting both nappes gives the two branches of a hyperbola; on one nappe you see one branch.

The cuts here avoid the cone’s vertex, so they are non-degenerate curves.

The source motivates cones with ice-cream and traffic cones, ellipses with planetary orbits, parabolas with a thrown ball or fountain jet, and hyperbolas with a lampshade’s light on a wall.

These are geometric models: projectile paths are approximately parabolic when drag is neglected, and the light pattern depends on the wall’s cut through the light cone.

The source also mentions Earth as an ellipsoid, the three-dimensional analogue of an ellipse.

To begin exploring the shapes, the Teacher copy suggests video prompt 1 and video prompt 2. The printed QR points to the conic visualization and applications resource. These are the source’s learning pointers; their external contents are not reproduced here. The original printed illustration credits the University of Georgia conics introduction; the figure below is independently reconstructed.

Open the 10 lesson diagrams
Open the chapter exploration

Worked example

For x2/9+y2/4=1x^2/9+y^2/4=1, the centre is (0,0)(0,0), horizontal intercepts are (±3,0)(\pm3,0) and vertical intercepts (0,±2)(0,\pm2). Solving for y produces two branches y=±21−x2/9y=\pm2\sqrt{1-x^2/9} for −3≤x≤3-3\leq x\leq3.

The denominator is the square of a semi-axis length, not the length itself.

Check your understanding

Find the asymptotes of x2/9−y2/4=1x^2/9-y^2/4=1.

Mastery learning objectives

At the end of this chapter, I should be able to:

General quadratic form

Rewrite Ax2+Bx+Cy2+Dy+E=0Ax^2+Bx+Cy^2+Dy+E=0 into a standard form: parabola, ellipse, circle or hyperbola.

Parabola: y=a(x−h)2+ky=a(x-h)^2+k

Identify line of symmetry x=hx=h.

Identify vertex/turning point (h,k)(h,k).

Sketch the graph, including any necessary features required by the question.

Parabola: (y−k)2=a(x−h)(y-k)^2=a(x-h)

Identify line of symmetry y=ky=k.

Identify vertex (h,k)(h,k).

Sketch the graph, including any necessary features required by the question.

Ellipse: (x−h)2/a2+(y−k)2/b2=1(x-h)^2/a^2+(y-k)^2/b^2=1

Identify centre (h,k)(h,k).

Identify length of its major axis.

Identify length of its minor axis.

Identify vertices (h−a,k)(h-a,k), (h+a,k)(h+a,k), (h,k−b)(h,k-b) and (h,k+b)(h,k+b).

Identify lines of symmetry x=hx=h and y=ky=k.

Sketch the graph, including any necessary features required by the question.

Circle: (x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2

Identify centre (h,k)(h,k).

Identify radius rr.

Identify lines of symmetry.

Sketch the graph, including any necessary features required by the question.

Horizontal hyperbola: (x−h)2/a2−(y−k)2/b2=1(x-h)^2/a^2-(y-k)^2/b^2=1

Identify centre (h,k)(h,k).

Identify vertices (h−a,k)(h-a,k) and (h+a,k)(h+a,k).

Identify lines of symmetry x=hx=h and y=ky=k.

Identify oblique asymptotes y=±(b/a)(x−h)+ky=\pm(b/a)(x-h)+k.

Sketch the graph, including any necessary features required by the question.

Vertical hyperbola: (y−k)2/b2−(x−h)2/a2=1(y-k)^2/b^2-(x-h)^2/a^2=1

Identify centre (h,k)(h,k).

Identify vertices (h,k−b)(h,k-b) and (h,k+b)(h,k+b).

Identify lines of symmetry x=hx=h and y=ky=k.

Identify oblique asymptotes y=±(b/a)(x−h)+ky=\pm(b/a)(x-h)+k.

Sketch the graph, including any necessary features required by the question.

Choose a practice question

The printed skill map (printed page 138) is retained below. A dash represents a blank source cell. Some Discussion subpart labels do not match the questions printed on the following page; the study links below use the actual questions.

Skill/conceptLecture examples, as printedDiscussion questions, as printedReview problems, as printed
Sketch parabola11(a)—
Sketch ellipse/circle41(b)(c), 2, 3 (1st part)5
Sketch hyperbola5, 71(d)(e)1, 6(a)
Sketch a conic on an existing curve to aid in solving the question—3 (last part)2(iii), 3, 4(i)
Using gradient concept to aid in solving the question—4(a)(iii)—
Prior topic: Graphing Techniques—4(a)(i)(ii)(b)2(i)(ii), 4(ii), 6(b)
Prior topic: Transformations——2(iv)

For the actual Discussion 1 equations, (a) and (b) are parabolas, (c) and (d) are ellipses, and (e) and (f) are hyperbolas. Discussion 3’s first graph is a rational curve, followed by an ellipse comparison. These qualifications preserve the printed map without directing you to the wrong conic.

Practice questions

11 examples and practice questions, in 2 sections.

More practice

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