Conic Section
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The document provides a detailed overview of conic sections, which are two-dimensional figures created by the intersection of a plane and a right circular cone. It outlines the four main types of conic sections: circles, parabolas, ellipses, and hyperbolas, including their definitions, properties, and equations. Additionally, examples are provided for each type, illustrating how to derive equations and sketch graphs.
Conic Section
- 1. CONIC SECTIONSPREPARED BY: ROQUI MABUGAY GONZAGA
- 2. When a straight line intersect a vertical line at the fixed point and rotate about the fixed point. The surface obtained is called a double right circular cone.
- 3. A double right circular cone consist of two cones joined at the fixed point called the vertex. A line that rotates about the vertex is called the generator. The line that remain fixed is called the axis. The right circular cone has a circular base and its axis is always perpendicular to its base. Vertex Generator Lower nappe Directrix Upper nappe Axis The Perimeter of its base is what we called the directrix. And the Lateral surface of the cone is called a nappe. A double right circular cone has two nappe, the cone above the vertex is the upper nappe and below the vertex is the lower nappe.
- 4. What is Conics? What is Conic Section?
- 5. What is Conic Section? two-dimensional figure created by the intersection of a plane and a right circular cone. If a plane intersects a right circular cone, we get two dimensional curves of different types. This curves is called the conic sections.
- 6. What is Conics? two-dimensional figure created by the intersection of a plane and a right circular cone. There are many types of curve produced when a plane slices through a cone. Parabola Circle Ellipse Hyperbola
- 7. Types of Conics Section 1. Circle Is made from a plane intersecting a cone parallel to its base. A circle is the locus of points that are equidistant from a fixed point (the center).
- 8. Types of Conics Section 1. Circle Let C be a given point. The set of all points P having the same distance from C is called a circle. The point C is called the center of the circle and the common distance is its radius. Q The circle has a center, C(0, 0) and radius r>0. A point P(x, y) is on the circle if and only if 𝑃𝐶 = 𝑟 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑥2 + 𝑦2 = 𝑟2 , r > 0 c P(x,y)
- 9. Types of Conics Section 1. Circle Let C be a given point. The set of all points P having the same distance from C is called a circle. The point C is called the center of the circle and the common distance is its radius. The circle has a center, C(h, k) and radius r>0. A point P(x, y) is on the circle if and only if 𝑃𝐶 = 𝑟 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (𝑥 − ℎ)2 +(𝑦 − 𝑘)2 = 𝑟2 , r > 0
- 10. Types of Conics Section Example 1:Write the equation of a circle which point (-6, 4) lies on the circle and center at (-5, 0). 1st Step: Find the radius (𝑥 − ℎ)2 +(𝑦 − 𝑘)2 = 𝑟2 center: (-5, 0), radius: (-6, 0) ---Apply Direct Substitution--- (𝑥 − ℎ)2 +(𝑦 − 𝑘)2 = 𝑟2 (−6 − (−5))2 +(4 − 0)2 = 𝑟2 (−6 + 5)2 +(4)2 = 𝑟2 (−1)2 +16 = 𝑟2 1+16 = 𝑟2 17= 𝑟2 2nd Step: Write the Standard equation (𝑥 − ℎ)2 +(𝑦 − 𝑘)2 = 𝑟2 (𝑥 + 5)2 +(𝑦 − 0)2 = 17 Or (𝑥 + 5)2 +𝑦2 = 17
- 11. Types of Conics Section 2. Parabola Is made from a plane intersecting a cone at an angle parallel to the slant edge.
- 12. Types of Conics Section 2. Parabola Let F be a given point, and ℓ a given line not containing F. The set of all points P such that its distances from F and from ℓ are the same, is called a parabola. Pℓ Directrix P(x, y) ℓ F
- 13. Types of Conics Section 2. Parabola A set of points on the coordinate plane that are of equal distance from a fixed point and line. The fixed point is called focus and the fixed line is called the directrix. Pℓ Directrix P(x, y) ℓ F
- 14. Types of Conics Section 2. Parabola The line connecting two points on the parabola and passing through the focus is called the latus rectum. The Axis of symmetry is the line which divides the parabola into two equal parts and passes through the vertex and the focus. Pℓ Directrix P(x, y) ℓ F
- 15. Types of Conics Section 2. Parabola Vertices Foci Directrices Equation Description 1. (ℎ, 𝑘) (ℎ ± 𝑝, 𝑘) 𝑥 = ℎ − 𝑝 (𝑦 − 𝑘)2= 4𝑝(𝑥 − ℎ) The axis of symmetry is 𝑦 = 𝑘 Open to the right if p>0 Open to the left if p<0 2. (ℎ, 𝑘) (ℎ, 𝑘 ± 𝑝) 𝑦 = 𝑘 − 𝑝 (𝑥 − ℎ)2= 4𝑝(𝑦 − 𝑘) The axis of symmetry is 𝑥 = ℎ Opens upward if p>0 Opens downward if p<0 Standard Equations of Parabola with vertex at (h, k) and axis of symmetry Parallel to a coordinate axis The general Equation of a parabola is; 𝐴𝑥2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0, 𝐸 ≠ 0 𝑖𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎 𝑜𝑝𝑒𝑛𝑠 𝑢𝑝𝑤𝑎𝑟𝑑/𝑑𝑜𝑤𝑛𝑤𝑎𝑟𝑑 𝐴𝑥2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0, 𝐷 ≠ 0 𝑖𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑎 𝑜𝑝𝑒𝑛𝑠 𝑠𝑖𝑑𝑒𝑤𝑎𝑦𝑠
- 16. Types of Conics Section Example 1: Identify the coordinates of the vertex, focus, and the equations of the axis of symmetry and directrix. Then Sketch the graph. 1. (𝑥 − 2)2 = 4 𝑦 − 1 opening of parabola: Upward p>0 4𝑝 = 4 4 = 4 P= 1 Vertex: (2, 1) Focus: (h, k+p)—(2, 1+1)—(2, 2) Axis of Symmetry: x=h—(x=2) Directrix: 𝑦 = 𝑘 − 𝑝 =1-1 y=0 h, k x=2 F (2, 2) V (2, 1) y = 0
- 17. Types of Conics Section 3. Ellipse Is made from a plane intersecting a cone at an angle parallel to the slant edge.
- 18. Types of Conics Section 3. Ellipse Let 𝐹1 and 𝐹2 be two distinct points. The set of all points P, Whose distances from 𝐹1 and from 𝐹2 add up to a certain constant, is called an ellipse. The points 𝐹1 and 𝐹2 are called the foci of the ellipse. 𝐹1 𝐹2 𝑃1 𝑃2 𝑷 𝟏 𝑭 𝟏 + 𝑭 𝟏 𝑷 𝟐 = 𝑷 𝟐 𝑭 𝟐 + 𝑭 𝟐 𝑷 𝟏
- 19. Types of Conics Section 3. Ellipse Let 𝐹1 and 𝐹2 be two distinct points. The set of all points P, Whose distances from 𝐹1 and from 𝐹2 add up to a certain constant, is called an ellipse. The points 𝐹1 and 𝐹2 are called the foci of the ellipse. Vertices Foci Endpoint of Minor Axis Equation Descriptio n Directrices Axis of symmetry (ℎ ± 𝑎, 𝑘) (ℎ ± 𝑐, 𝑘) (ℎ ± 𝑏, 𝑘) (𝑥 − ℎ)2 𝑎2 + (𝑦 − 𝑘)2 𝑏2 a>b Major Axis is Horizontal 𝑥 = ℎ ± 𝑎 𝑒 Both Axis (ℎ, 𝑘 ± 𝑎) (ℎ, 𝑘 ± 𝑐) (ℎ, 𝑘 ± 𝑏) (𝑥 − ℎ)2 𝑏2 + (𝑦 − 𝑘)2 𝑎2 Major Axis is Vertical 𝑦 = 𝑘 ± 𝑎 𝑒 Both Axis Standard Equation of Ellipse with Center (h, k)
- 20. Types of Conics Section 3. Ellipse Eccentricity: 𝑒 = 𝑐 𝑎 e(a constant)= 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑓𝑜𝑐𝑢𝑠 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑜 𝑑𝑖𝑟𝑒𝑐𝑡𝑟𝑖𝑥 e=0 for Circle 0<e<1 for Ellipse e=1 for Parabola e>1 for hyperbola The general form of the equation of an ellipse is 𝐴𝑥2 + 𝐶𝑦2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 with AC>0 and a≠ 0. Properties: • a, b, c • Center • Vertex • Covertex( endpoints of Minor axis) • Foci • Directrix • Axis of Symmetry • Major Axis • Length of Major Axis • Minor Axis • Length of Minor Axis
- 21. Types of Conics Section Example 1: Identify the properties of the equations and sketch the graph. 1. (𝑥−2)2 16 + (𝑦−1)2 4 = 1 a=4, b=2, c= 12 Center(h, k): 2, 1) Vertex (h±𝑎, 𝑘): (2 ±4, 1) 𝑉1: 6, 1 & 𝑉2: −2, 1 Foci (h ±c, k): (2± 12, 1) 𝐹1: 2 + 12, 1 or (5.46, 1) & 𝐹2: 2 − 12, 1 or −1.46, 1 Covertex (h, k±𝑏): (2, 1±2) 𝐵1: 2, 3 & 𝐵2: 2, −1 Directrix 𝑥 = ℎ ± 𝑎 𝑒 : 𝑋1: 2 + 4 12 4 ; 2+4.61=6.61 𝑋2: 2 − 4 12 4 ; 2-4.61=-2.61 Axis of Symmetry: x=2; y=1 Major Axis: Horizontal Minor Axis: Vertical 𝐹1 𝐹2 c 𝐵1 𝐵2 𝑉1 𝑉2
- 22. Types of Conics Section 4. Hyperbola Is made from a plane intersecting both halves of a double cone, but not passing through the apex.
- 23. Types of Conics Section 4. Hyperbola Let 𝐹1 and 𝐹2 be two distinct points. The set of all points P, Whose distances from 𝐹1 and 𝐹2 differ by a certain constant, is called a hyperbola. The points 𝐹1 and 𝐹2 are called the foci of the hyperbola. 𝐹1 𝐹2 𝑃1 𝑃2 𝑭 𝟏 𝑷 𝟏 − 𝑭 𝟐 𝑷 𝟐 = 𝑭 𝟏 𝑷 𝟐 − 𝑭 𝟐 𝑷 𝟐
- 24. Types of Conics Section 4. Hyperbola A hyperbola is a set of all points in the plane such that the absolute value of the difference of the distances from two fixed points are called the foci of the hyperbola. 𝐹1 𝐹2 𝑃1 𝑃2 𝑭 𝟏 𝑷 𝟏 − 𝑭 𝟐 𝑷 𝟐 = 𝑭 𝟏 𝑷 𝟐 − 𝑭 𝟐 𝑷 𝟐
- 25. Types of Conics Section 4. Hyperbola Transverse Axis of the parabola is the line that connects the vertices and has the length of 2a. Conjugate Axis is the line that connects the co-vertices and has a length of 2b 𝐹1 𝐹2 𝑃1 𝑃2 𝑭 𝟏 𝑷 𝟏 − 𝑭 𝟐 𝑷 𝟐 = 𝑭 𝟏 𝑷 𝟐 − 𝑭 𝟐 𝑷 𝟐
- 26. Types of Conics Section 4. Hyperbola Equation Vertices Foci Endpoints of conjugate axis Asymptotes Directrices (𝑥 − ℎ)2 𝑎2 − 𝑦 − 𝑘 2 𝑏2 = 1 𝑏2 = 𝑐2 − 𝑎2 𝑐2 = 𝑎2 + 𝑏2 (ℎ ± 𝑎, 𝑘) (ℎ ± 𝑐, 𝑘) (ℎ, 𝑘 ± 𝑏) 𝑦 − 𝑘 = ± 𝑏 𝑎 (𝑥 − ℎ) 𝑥 = ℎ ± 𝑎 𝑒 (𝑦 − 𝑘)2 𝑎2 − 𝑥 − ℎ 2 𝑏2 = 1 𝑏2 = 𝑐2 − 𝑎2 (ℎ, 𝑘 ± 𝑎) (ℎ, 𝑘 ± 𝑐) (ℎ ± 𝑏, 𝑘) 𝑦 − 𝑘 = ± 𝑎 𝑏 (𝑥 − ℎ) 𝑦 = 𝑘 ± 𝑎 𝑒 Standard Equation of Ellipse with Center (h, k)
- 27. Types of Conics Section 4. Hyperbola Steps in graphing the Hyperbola: 1. Find the vertices of the hyperbola 2. Draw the fundamental triangle 3. Sketch the asymptote as diagonals 4. Sketch the graph. Each graph goes through the vertex and approaches each asymptotes.
- 28. Types of Conics Section Example 1: Sketch the graph: 𝑥2 52 − 𝑦2 42=1 Center: (0, 0) Vertex (h±𝑎, 𝑘)= (0±5, 0) 𝑉1: 5, 0 & 𝑉2: −5, 0 F(h±𝑐, 𝑘): F(0± 41, 0) 𝐹1: 41 ,0 & 𝐹2: − 41, 0 B(h, k±𝑏): (0, 0±4) 𝐵1: 0, 4 & 𝐵2: 0, −4 Assymptotes: 𝑦 − 𝑘 = ± 𝑏 𝑎 𝑥 − ℎ 𝑦 − 0 = ± 4 5 (𝑥 − 0) 𝑦1= 4 5 𝑥 & 𝑦2=- 4 5 𝑥 h , k 𝑎2 = 52 a=5 𝑏2 = 42 𝑏 = 4 𝑐2 = 𝑎2 + 𝑏2 𝑐2 = 25 + 16 → 𝑐 = 41 Directrix: x=ℎ ± 𝑎 𝑒 e = 𝑐 𝑎 = 41 5 x=0 ± 5 41 5 𝑥1 = 3.9 𝑥2 = −3.9
- 29. Types of Conics Section Graph: 𝑥2 52 − 𝑦2 42=1 𝑦2=- 4 5 𝑥 𝑦1= 4 5 𝑥 c 𝐹1𝐹2 𝐵1 𝐵2 𝑉1𝑉2 y x