Conic Sections 2 - Chapter 3 - Conic Sections 2 Cheat Sheet

Q: How are conic sections defined in this chapter?

Conic sections are formed by slicing two cones with a plane. The intersection curves of cones and planes produce the conic sections discussed.

Q: What are the standard Cartesian and parametric forms of an ellipse?

An ellipse has a Cartesian equation of the form shown in the sheet and can also be expressed parametrically with x and y written using cosine and sine. Parameters are identified from the general ellipse form.

Q: How does eccentricity determine whether a conic is an ellipse, parabola, or hyperbola?

Eccentricity is defined as the ratio of the distance from the focus to the shortest distance from the directrix. For 0 1 corresponds to a hyperbola.

Oct 14, 2026

Upload by James

Conic Sections 2 Cheat Sheet explains how conic sections arise from slicing two cones and how key curves—circles, ellipses, parabolas, and hyperbolas—are represented in modelling. The sheet summarizes ellipse Cartesian and parametric forms, including how to determine parameters from the standard equation. It also covers hyperbolas with Cartesian equations, asymptotes, and parametric definitions using hyperbolic trigonometric functions. Finally, it introduces eccentricity via focus-directrix ratios and links eccentricity ranges to ellipse, parabola, and hyperbola classifications.

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Ellipses

Hyperbolas

Eccentricity

Conic Sections 2 Cheat Sheet This chapter aims to build upon the graph sketching skills you developed in year 1 and the content covered in the previous chapter, conic sections 1. As described by the name, conic sections are planes produced by slicing two cones placed point to point-more mathematically, the intersection of cones with planes are the graphs that you will see in this chapter. Conic sections are used in modelling: planets travel in ellipses, the mirrors in solar power stations are parabolas and a variety of conic sections are used in engineering. In the set,of diagrams to the right, diagram 1 shows how a circle is obtained from a conic section, 2 shows an ellipse, 3 shows a parabola and 4 shows a hyperbola. Ellipses If you slice a cone such to produce a closed curve, the curve you have obtained is called an ellipse-a circle is a special case of an ellipse. • An ellipse has the Cartesian equation 􀯫􀯔22 + 􀯬􀯕22 = 1, where 􀜽 and substituting in 􀝔 = 0 that the y-intercepts are at 􀝕 = ±􀜾 , and by substituting in 􀝕 = 0 the x-intercepts are at 􀝔 = ±􀜽 . • Ellipses can also be defined using parametric equations. An ellipse has parametric equations 􀝔 = 􀜽cos 􀝐, 􀝕 = 􀜾sin 􀝐, 0 ≤ 􀝐 < 2􀟨 . This can be derived by substituting 􀝔 = 􀜽cos 􀝐 and 􀝕 = 􀜾sin 􀝐 into the Cartesian equation. A general point P on an ellipse has co-ordinates (􀜽cos 􀝐, 􀜾sin 􀝐). 􀜾 are constants. It is simple to see by Example 1: Find the parametric equations for the ellipse with Cartesian co-ordinates 9􀝔 2 + 25􀝕2 = 225 | Rewrite the equation given in the question to the general form for ellipses. | 9􀝔 2 + 25􀝕2 = 225
2 2
􀝔25 + 􀝕9~~ ~~ = 1 | | --- | --- | | From the general form, identify 􀜽 and 􀜾 and use this to find the parametric forms. | 􀜽 = 5, 􀜾 = 3
So the parametric equations are
􀝔 = 5cos 􀝐, 􀝕 = 3sin 􀝐 | Again, lots of questions in exams will build on pervious knowledge- make sure you are confident with finding points of intersections and distances! Hyperbolas We have already seen rectangular hyperbolas in the previous chapter, conic sections 1. However, it isn’t necessary for hyperbolas to have perpendicular asymptotes, and we can find both Cartesian and parametric equations for standard hyperbolas • A standard hyperbola has Cartesian equation 􀯫􀯔22 − 􀯬􀯕22 = 1 . Note that the fractions are not added-don’t get them confused with ellipses! By substituting in 􀝕 = 0, we can see that 􀝔 2 = 􀜽2 . To investigate the asymptotes, we can look at see what happens to the equation as 􀝔 and 􀝕 tend to infinity. As they tend to infinity, 􀯫􀯔22 ≈ 􀯬􀯕22, and thus the asymptotes are 􀝕 = ± 􀯕􀯔 􀝔 . The equations of the asymptotes can also be found in the formula booklet. These asymptotes are perpendicular to one another, however, don’t confuse all hyperbolas with perpendicular asymptotes as rectangular-only the curves of the form 􀝔􀝕 = 􀜿 2 are. • The parametric equations of the hyperbolas are connected to the hyperbolic trigonometric functions and their identities. The standard hyperbola has parametric equations 􀝔 = ±􀜽cosh 􀝐, 􀝕 = 􀜾sinh 􀝐, 􀝐 ∈ℝ, or equivalently, 􀝔 = 􀜽sec 􀟠 , 􀝕 = 􀜾tan 􀟠, −􀟨 ≤ 􀟠 < 􀟨, 􀟠 ≠ ± 􀰗2 . Example 2: Find the Cartesian equation of the hyperbola given by 􀝔 = ±5cosh 􀝐, 􀝕 = 3sinh 􀝐 . | Use the hyperbolic trigonometric identity cosh2 􀟠 − sinh2􀟠 = 1 to eliminate the parameter 􀝐 | 􀝔 2 = 25cosh2 􀝐
2 = 9sinh2􀝐
􀝕
2 2
􀝔25 − 􀝕9~~ ~~ = cosh2 􀝐 − sinh2􀝐
2 2
􀝔25 − 􀝕9~~ ~~ = 1 | | --- | --- | Eccentricity Like shown with the parabola in the previous topic, conic sections can be defined in terms of their focus-directrix properties. To do this we need to consider the eccentricity of a conic section. • For all points, P, on a conic section, the eccentricity of the curve is defined as the ratio of the distance of P from the focus, denoted S, to the shortest distance from P to the directrix (the point on the directrix is denoted M) . The ratio is denoted 􀝁 and is a unique non-negative real number that characteri

FAQs

How are conic sections defined in this chapter?

Conic sections are formed by slicing two cones with a plane. The intersection curves of cones and planes produce the conic sections discussed.

What are the standard Cartesian and parametric forms of an ellipse?

An ellipse has a Cartesian equation of the form shown in the sheet and can also be expressed parametrically with x and y written using cosine and sine. Parameters are identified from the general ellipse form.

How does eccentricity determine whether a conic is an ellipse, parabola, or hyperbola?

Eccentricity is defined as the ratio of the distance from the focus to the shortest distance from the directrix. For 0 < e < 1 the point is on an ellipse, e = 1 corresponds to a parabola, and e > 1 corresponds to a hyperbola.

Oct 14, 2026

Upload by James