Precalculus 2nd quarter (CONIC SECTIONS).pptx

This is a topic in precalculus first quarter

1. CONIC SECTIONS SUBJECT: PRE – CALCULUS SUBJECT TEACHER: JOSEPHINE E. PATAC

2. PRE-TEST: Fill in the blank with appropriate word or phrase.

3. TYPES OF CONIC SECTIONS LESSON 1

4. WHAT HAPPENED In mathematics, a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane. The conic sections were named and studied as long ago as 200 BC when Apollonius of Perga undertook a systematic study of their properties.

5. What You Need to Know Conic sections (or conics) are a particular class of curves which oftentimes appear in nature and which have applications in other fields. One of the first shapes we learned, a circle, is a conic. When you throw a ball, the trajectory it takes is a parabola. The orbit taken by each planet around the sun is an ellipse. Properties of hyperbolas have been used in the design of certain telescopes and navigation systems.

6. Key Points! A conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane; the three types are parabolas, ellipses, and hyperbolas.

7. The following key terms must be familiarized in our discussion about conic sections. ❖ Vertex is an extreme point on a conic section. ❖ Asymptote is a straight line which a curve approaches arbitrarily closely as it goes to infinity

8. ❖ Focus is a point used to construct and define a conic section, at which rays reflected from the curve converge (plural: foci). ❖ Nappe is one half of a double cone. ❖ Directrix is a line used to construct and define a conic section; a parabola has one directrix; ellipses and hyperbolas have two (plural: directrices).

9. ▪ Every conic section has certain features, including at least one focus and directrix . Parabolas have one focus and directrix , while ellipses and hyperbolas have two of each. ▪ The circle is type of ellipse, and is sometimes considered to be a fourth type of conic section.

10. The illustration below shows a cone and conic sections. The nappes and the four conic sections. Each conic is determined by the angle the plane makes with the axis of the cone.

11. Common Parts of Conic Sections A focus is a point about which the conic section is constructed. In other words, it is a point about which rays reflected from the curve converge. A directrix is a line used to construct and define a conic section. The distance of a directrix from a point on the conic section has a constant ratio to the distance from that point to the focus. As with the focus, a parabola has one directrix , while ellipses and hyperbolas have two .

12. Let us now discuss the different types of conic sections. ➢ Parabola - A parabola is the set of all points whose distance from a fixed point, called the focus, is equal to the distance from a fixed line, called the directrix. The point halfway between the focus and the directrix is called the vertex of the parabola.

13. ➢ Ellipses An ellipse is the set of all points for which the sum of the distances from two fixed points (the foci) is constant. In the case of an ellipse, there are two foci, and two directrices. On the right is a typical ellipse graphed as it appears on the coordinate plane.

14. ➢ Hyperbolas A hyperbola is the set of all points where the difference between their distances from two fixed points (the foci) is constant. In the case of a hyperbola, there are two foci and two directrices. Hyperbolas also have two asymptotes. On the right is a graph of a typical hyperbola.

15. ➢ Degenerate conics A degenerate conic is formed when the plane does not pass through the vertex. It could be a point, a line, or two intersecting lines.

16. circle Lesson 2

17. Definition of a 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 its radius.

18. FINDING THE STANDARD FORM OF THE EQUATION OF A CIRCLE

19. POST TEST: I. Label the different types of conic sections and its parts as numbered below.

20. II. Fill in the blanks. 1. A set of all points for which the sum of the distances from two fixed points (the foci) is called _______. 2. An ellipse has ______ number of focus/ foci. 3. An ellipse has ______ directrix/directrices. 4. Aside from the foci and directrices, a hyperbola also has two __________. 5. A _________ is the set of all points where the difference between their distances from two fixed points (the foci) is constant. 6. In a parabola, the point half-way between the focus and the directrix is called _________. 7. A set set of all points whose distance from a fixed point, called the focus, is equal to the distance from a fixed line, called directrix is what we call as _________.

21. I II . Find the equation of the circle with the following conditions. 1. Center at the origin, radius 4. 2. Center at (2,-3), radius = 5 3. Center at (-2,-3), and passes through (-2,0). 4. Center at (-3,5), diameter 12. 5. Given the standard form of the equation of a circle ( 𝑥 − 7) ² + ( 𝑦 – 8) ² = 144, find the center and the radius.