Lesson Notes By Weeks and Term v3 - Senior Secondary 3
Conic sections
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Conic sections
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Subject: Further Mathematics
Class: Senior Secondary 3
Term: 1st Term
Week: 3
Theme: Coordinate Geometry
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Know the equations of parabola, ellipse and hyperbola in a rectangular cartesian coordinates and parametric equations. To recognize practical solid shapes of parabolic, elliptic and hyperbolic types.
directrix is `x = -a`. `x = -2` ---
B. The Ellipse Definition: An ellipse is the locus of a point that moves such that the sum of its distances from two fixed points (foci, `F1` and `F2`) is constant. This constant sum is equal to the length of the major axis, `2a`.
Key Features: Center: The midpoint of the segment connecting the two foci. Foci (`F1, F2`): Two fixed points from which the ellipse is defined. Vertices (`V1, V2`): The endpoints of the major axis. Co-vertices (`B1, B2`): The endpoints of the minor axis.
Major Axis: The longer axis of the ellipse, passing through the foci and vertices. Length = `2a`.
Minor Axis: The shorter axis of the ellipse, perpendicular to the major axis and passing through the center. Length = `2b`.
Eccentricity (e): `e = c/a`, where `c` is the distance from the center to a focus. For an ellipse, `0 b`)
Vertices: `(±a,0)` Foci: `(±c,0)` where `c = √(a2 - b2)` Co-vertices: `(0,±b)` Directrices: `x = ±a/e`
2. Vertical Major Axis (foci on y-axis): `x2/b2 + y2/a2 = 1` (where `a > b`)
Vertices: `(0,±a)` Foci: `(0,±c)` where `c = √(a2 - b2)` Co-vertices: `(±b,0)` Directrices: `y = ±a/e` General Form (Center at `(h,k)`):
1. Horizontal Major Axis: `(x-h)2/a2 + (y-k)2/b2 = 1`
2. Vertical Major Axis: `(x-h)2/b2 + (y-k)2/a2 = 1` Parametric Equations for `x2/a2 + y2/b2 = 1`: `x = a cosθ` `y = b sinθ` where `θ` is the parameter.
Worked Example 2 (Ellipse): An architect designs an elliptical archway for a building in Abuja. The arch has a major axis of 10 meters and a minor axis of 6 meters, centered at the origin. Find the equation of the ellipse and the coordinates of its foci. Assume the major axis is horizontal.
Solution:
1. Identify parameters: Major axis = `2a = 10` meters => `a = 5` meters. Minor axis = `2b = 6` meters => `b = 3` meters. Since the major axis is horizontal and the center is at the origin, the standard equation is `x2/a2 + y2/b2 = 1`.
2. Equation of the ellipse: `x2/52 + y2/32 = 1` `x2/25 + y2/9 = 1`
3. Find 'c' for foci: Use `c2 = a2 - b2`. `c2 = 52 - 32 = 25 - 9 = 16` `c = √16 = 4`
4. Coordinates of the foci: Since the major axis is horizontal, the foci are `(±c,0)`. Foci are `(4,0)` and `(-4,0)`. ---
C. The Hyperbola Definition: A hyperbola is the locus of a point that moves such that the absolute difference of its distances from two fixed points (foci, `F1` and `F2`) is constant. This constant difference is equal to the length of the transverse axis, `2a`.
Key Features: Center: The midpoint of the segment connecting the two foci. Foci (`F1, F2`): Two fixed points from which the hyperbola is defined. Vertices (`V1, V2`): The endpoints of the transverse axis.
Transverse Axis: The axis that passes through the foci and vertices. Length = `2a`.
Conjugate Axis: Perpendicular to the transverse axis, passing through the center. Length = `2b`.
Eccentricity (e): `e = c/a`, where `c` is the distance from the center to a focus. For a hyperbola, `e > 1`. Relationship between a, b, c: `c2 = a2 + b2` or `b2 = a2(e2 - 1)`.
Asymptotes: Two lines that the hyperbola branches approach but never touch as they extend infinitely.
Directrices: `x = ±a/e` (for horizontal transverse axis) or `y = ±a/e` (for vertical transverse axis). Standard Equations (Center at `(0,0)`):
1. Horizontal Transverse Axis (foci on x-axis): `x2/a2 - y2/b2 = 1` Vertices: `(±a,0)` Foci: `(±c,0)` where `c = √(a2 + b2)` Asymptotes: `y = ±(b/a)x`
2. Vertical Transverse Axis (foci on y-axis): `y2/a2 - x2/b2 = 1` Vertices: `(0,±a)` Foci: `(0,±c)` where `c = √(a2 + b2)` * Asymptotes: `y = ±(a/b)x` General Form (Center at `(h,k)`):
1. Horizontal Transverse Axis: `(x-h)2/a2 - (y-k)2/b2 = 1`
2. Introduction to Conic Sections Conic sections are curves formed by the intersection of a plane with a double-napped right circular cone. The angle at which the plane intersects the cone determines the type of conic section produced.
Circle: Formed when the plane intersects the cone horizontally, perpendicular to the axis of the cone. (Often considered a special case of the ellipse, with eccentricity e=0).
Ellipse: Formed when the plane intersects one nappe of the cone at an angle to the axis, not parallel to a generator.
Parabola: Formed when the plane intersects one nappe of the cone parallel to a generator (slant edge) of the cone.
Hyperbola: Formed when the plane intersects both nappes of the cone.
Eccentricity (e): A Unifying Property A conic section can also be defined as the locus of a point `P` such that its distance from a fixed point `F` (the focus) bears a constant ratio to its distance from a fixed line `L` (the directrix). This constant ratio is called the eccentricity, denoted by `e`.
Parabola: `e = 1` (distance from focus = distance from directrix)
Ellipse: `0 1` (distance from focus > distance from directrix) ---
A. The Parabola Definition: A parabola is the locus of a point that moves such that its distance from a fixed point (the focus, `F`) is equal to its distance from a fixed straight line (the directrix, `L`).
Key Features: Vertex (V): The turning point of the parabola, equidistant from the focus and directrix.
Focus (F): A fixed point from which the parabola is defined.
Directrix (L): A fixed line from which the parabola is defined.
Axis of Symmetry: The line passing through the vertex and the focus, perpendicular to the directrix.
Latus Rectum: A chord passing through the focus and perpendicular to the axis of symmetry. Its length is `4a`.
Standard Equations: For a parabola with vertex at the origin `(0,0)`:
1. Opens Right: `y2 = 4ax` Vertex: `(0,0)` Focus: `(a,0)` Directrix: `x = -a` Axis of symmetry: `y = 0` (x-axis)
2. Opens Left: `y2 = -4ax` Vertex: `(0,0)` Focus: `(-a,0)` Directrix: `x = a` Axis of symmetry: `y = 0` (x-axis)
3. Opens Upwards: `x2 = 4ay` Vertex: `(0,0)` Focus: `(0,a)` Directrix: `y = -a` Axis of symmetry: `x = 0` (y-axis)
4. Opens Downwards: `x2 = -4ay` Vertex: `(0,0)` Focus: `(0,-a)` Directrix: `y = a` Axis of symmetry: `x = 0` (y-axis) General Form (Vertex at `(h,k)`):
1. Horizontal axis: `(y-k)2 = 4a(x-h)` (opens right) or `(y-k)2 = -4a(x-h)` (opens left)
2. Vertical axis: `(x-h)2 = 4a(y-k)` (opens up) or `(x-h)2 = -4a(y-k)` (opens down) Parametric Equations for `y2 = 4ax`: `x = at2` `y = 2at` where `t` is the parameter.
Worked Example 1 (Parabola): A parabolic reflector for a satellite dish, like those used for DSTV in Nigerian homes, has its vertex at the origin and its axis of symmetry along the x-axis. If the focus is at `(2,0)`, find the equation of the parabola and the equation of its directrix.
Solution:
1. Identify the type: The focus is `(2,0)`, which means the parabola opens to the right. The standard form is `y2 = 4ax`.
2. Determine 'a': For `y2 = 4ax`, the focus is `(a,0)`. Comparing `(a,0)` with `(2,0)`, we get `a = 2`.
3. Equation of the parabola: Substitute `a = 2` into `y2 = 4ax`: `y2 = 4(2)x` `y2 = 8x`
4. Equation of the directrix: For `y2 = 4ax`, the directrix is `x = -a`. `x = -2` ---
B. The Ellipse Definition: An ellipse is the locus of a point that moves such that the sum of its distances from two fixed points (foci, `F1` and `F2`) is constant. This constant sum is equal to the length of the major axis, `2a`.
Key Features: Center: The midpoint of the segment connecting the two foci. Foci (`F1, F2`): Two fixed points from which the ellipse is defined. Vertices (`V1, V2`): The endpoints of the major axis. * *Co-vertices (`B1, = ±a/e` (for horizontal transverse axis) or `y = ±a/e` (for vertical transverse axis). Standard Equations (Center at `(0,0)`):
1. Horizontal Transverse Axis (foci on x-axis): `x2/a2 - y2/b2 = 1` Vertices: `(±a,0)` Foci: `(±c,0)` where `c = √(a2 + b2)` Asymptotes: `y = ±(b/a)x`
2. Vertical Transverse Axis (foci on y-axis): `y2/a2 - x2/b2 = 1` Vertices: `(0,±a)` Foci: `(0,±c)` where `c = √(a2 + b2)` Asymptotes: `y = ±(a/b)x` General Form (Center at `(h,k)`):
1. Horizontal Transverse Axis: `(x-h)2/a2 - (y-k)2/b2 = 1`
2. Vertical Transverse Axis: `(y-k)2/a2 - (x-h)2/b2 = 1` Parametric Equations for `x2/a2 - y2/b2 = 1`: `x = a secθ` `y = b tanθ` where `θ` is the parameter. (Alternatively: `x = a cosh t`, `y = b sinh t`)
Worked Example 3 (Hyperbola): A hyperbola has its foci at `(±5,0)` and its vertices at `(±3,0)`. Find the equation of the hyperbola and the equations of its asymptotes.
Solution:
1. Identify parameters: Foci are `(±5,0)`, so `c = 5`. The transverse axis is horizontal. Vertices are `(±3,0)`, so `a = 3`. The standard equation for a horizontal transverse axis is `x2/a2 - y2/b2 = 1`.
2. Find 'b': Use `c2 = a2 + b2`. `52 = 32 + b2` `25 = 9 + b2` `b2 = 25 - 9 = 16` `b = √16 = 4`
3. Equation of the hyperbola: Substitute `a = 3` and `b = 4` into the standard equation: `x2/32 - y2/42 = 1` `x2/9 - y2/16 = 1`
4. Equations of the asymptotes: For a horizontal transverse axis, the asymptotes are `y = ±(b/a)x`. `y = ±(4/3)x` --- Recognizing practical solid shapes: Parabolic shapes: Satellite dishes, car headlight reflectors, solar cookers, suspension bridge cables (e.g., Lekki-Ikoyi Link Bridge if approximated as parabolic), the path of a stone thrown in the air.
Elliptic shapes: Orbits of planets around the sun, whispering galleries (architectural designs allowing sound to be heard clearly from specific points), some architectural arches or domes, the cross-section of a long loaf of bread or certain fruits.
Hyperbolic shapes: Designs of cooling towers for power plants, specific gears in machinery, path of a comet entering the solar system (sometimes hyperbolic), principles in LORAN navigation. Introduction (10 minutes)
Teacher Activity: Begin by asking students what shapes they recognize in everyday objects like a satellite dish, a car headlight, or a sports field. Display pictures of these objects.
Teacher Activity: Introduce the concept of conic sections by explaining that these familiar shapes (parabola, ellipse) are formed by cutting a cone with a plane at different angles. Use a visual aid (a physical cone model if available, or diagrams) to demonstrate this.
Student Activity: Students share their observations of real-life shapes and engage in a brief discussion about where these shapes might be found locally. Development of Key Concepts (40 minutes)
A. Parabola (15 minutes)
Teacher Activity: Define a parabola using the focus-directrix property. Present the standard equation `y2 = 4ax` (and its variations) for a parabola with vertex at the origin. Clearly define `a`, vertex, focus, directrix, and axis of symmetry. Illustrate with a diagram. Explain how to identify these features from the equation. Introduce the parametric equations `x = at2`, `y = 2at`. Work through Worked Example 1, emphasizing step-by-step reasoning.
Student Activity: Students take notes, sketch diagrams, and ask clarifying questions. Students attempt to identify the features of a given parabolic equation. Students follow along and participate in solving Worked Example
1. B.
Ellipse (15 minutes)
Teacher Activity: Define an ellipse using the two-foci property. Present the standard equations `x2/a2 + y2/b2 = 1` and `x2/b2 + y2/a2 = 1` (with `a > b`) for an ellipse with center at the origin. Clearly define `a`, `b`, foci, vertices, major/minor axes, and eccentricity `e`. Illustrate with a diagram. Explain the relationship `c2 = a2 - b2` and how to determine the orientation (horizontal/vertical major axis). Introduce the parametric equations `x = a cosθ`, `y = b sinθ`. Work through Worked Example
2. Student Activity: Students take notes, sketch diagrams, and ask clarifying questions. Students identify the characteristics and orientation of given elliptical equations. Students follow along and participate in solving Worked Example
2. C.
Hyperbola (10 minutes)
Teacher Activity: Define a hyperbola using the two-foci property (difference of distances). Present the standard equations `x2/a2 - y2/b2 = 1` and `y2/a2 - x2/b2 = 1` for a hyperbola with center at the origin. Clearly define `a`, `b`, foci, vertices, transverse/conjugate axes, and eccentricity `e`. Illustrate with a diagram, including asymptotes. Explain the relationship `c2 = a2 + b2` and how to determine the orientation (horizontal/vertical transverse axis). Introduce the parametric equations `x = a secθ`, `y = b tanθ`. Work through Worked Example
3. Student Activity: Students take notes, sketch diagrams, and ask clarifying questions. Students identify the characteristics and orientation of given hyperbolic equations. Students follow along and participate in solving Worked Example
3. Consolidation and Recognition of Shapes (10 minutes)
Teacher Activity: Review all three conic sections, briefly comparing their definitions and equations. Present a series of images of real-world objects and ask students to identify which conic section best describes their shape (e.g., satellite dish, a bridge arch, an oval running track).
Student Activity: Students identify the conic sections from the images and explain their reasoning.
Satellite Dishes and Headlights (Parabola): In many Nigerian homes and businesses, satellite dishes (e.g., DSTV, StarTimes) are ubiquitous. These dishes are parabolic reflectors designed to focus parallel incoming signals (from satellites) onto a single point (the LNB - Low Noise Block converter) or to beam signals outwards from a source at the focus. Car headlights and solar cookers also use parabolic mirrors for similar light-focusing or radiating properties. This helps students understand the practical utility of the focus-directrix property. Bridge Arches and Architectural Designs (Parabola & Ellipse): Several bridges across Nigeria, like pedestrian bridges or smaller vehicle bridges, incorporate arch designs. While some are circular, many are parabolic or elliptical, providing structural strength and aesthetic appeal. For example, a discussion can be had about how the Lekki-Ikoyi Link Bridge's cable design, while catenary in strict mathematical terms, can be approximated to a parabola for simpler analysis in certain contexts. Elliptical shapes are also used in the design of certain architectural elements, such as windows, domes, and decorative patterns in public buildings or private residences.
Planetary Orbits and Acoustics (Ellipse): Though not directly observable in a Nigerian context, the elliptical orbits of planets around the Sun (and moons around planets) are a fundamental application of ellipses in astrophysics. This can be related to the Nigerian Space Research and Development Agency (NASRDA) and its aspirations. Additionally, "whispering galleries" – acoustically designed rooms where a whisper at one focus can be heard clearly at another focus – demonstrate the reflective property of ellipses. While there might not be a famous one in Nigeria, the principle is used in designing auditoriums, lecture halls (like those in universities such such as UNILAG or ABU), and public spaces to optimize sound distribution.