conic_sections_and_circles.pptxpptppptppppppt
- 2. Session Objectives For this two-hour period, SHS students in Precalculus are expected to develop the following learning competencies: 1. illustrate the different types of conic sections: parabola, ellipse, circle, hyperbola, and degenerate cases; 2. define a circle; 3. determine the standard form and general form of the equation of a circle; and 4. graph a circle in a rectangular coordinate system.
- 3. Conic Sections Intersections of cones and planes 1
- 4. Starting with Basics Explain the following mathematical terms: ◉ Undefined Terms in Geometry ◉ Intersecting lines ◉ Parallel lines ◉ Perpendicular lines ◉ Angle ◉ Cone
- 5. What are CONIC SECTIONS? If a plane intersects a double right circular cones, the intersection is a two-dimensional curve of different types. These curves are called conic sections. Also called conics, these different types are: CIRCLE ELLIPSE PARABOLA HYPERBOLA
- 6. What are CONIC SECTIONS? CIRCLE ELLIPSE PARABOLA HYPERBOLA
- 8. CIRCLES A circle is the set of all points equidistant to a fixed point. radiu s center
- 9. EQUATION OF A CIRLE Consider a circle plotted on the Cartesian Coordinate plane. Determine the coordinates of the center and its radius.
- 10. EQUATION OF A CIRLE Assume that we do not know the actual coordinates of the center. What if the center is at the point of origin? 𝐶 ( h , 𝑘) 𝑃 ( 𝑥 , 𝑦)
- 11. STANDARD FORM of the equation of a circle The standard form of the equation of a circle whose center is at the origin with radius, r, is For any circle whose center have coordinates (h, k) with radius, r, the standard form of its equation is
- 12. STANDARD FORM of the equation of a circle Give the standard form of the equation of the circle described in each item below. 1. Center at the origin, 2. and r = 3. and r = 10
- 13. EQUATION OF A CIRLE Solve for the equation of the following circles: ◉ circle A ◉ circle B ◉ center (7, 2), r= 8 − ◉ center (-8,0 ) r= 11
- 14. Check your Understanding Find the center and radius of the given circle. 1. + =100 2. + = 3. + =16 4. = 10 5. =121
- 15. GENERAL FORM of the equation of a circle The general form of the equation of a circle is where
- 16. GENERAL FORM of the equation of a circle Transform the following equations into the general form:
- 17. Transforming equations Transform the following equations into the standard form:
- 18. Transforming equations Group the terms based on the variables. Complete the square to make each group of terms a perfect square trinomial (PST). Add the constants of the PSTs to the right side. Express the PSTs as squares of
Editor's Notes
- #4: The three undefined terms in Geometry are the point, line, and plane. A point is represented by a dot, a line is a set of points that form a straight line, and a plane is a perfectly flat surface extending infinitely in every direction. Intersecting lines are two or more lines that meet at exactly one point. Parallel lines are lines that do not intersect, even if extended infinitely in both directions. Perpendicular lines are special type of intersecting lines that intersect to form a right angle. An angle is composed of two rays with a common endpoint. A cone is a geometric solid composed of a point called the vertex and a circular base.
- #6: Circle - when the plane is horizontal Ellipse - when the (tilted) plane intersects only one cone to form a bounded curve Parabola - when the plane intersects only one cone to form an unbounded curve Hyperbola - when the plane (not necessarily vertical) intersects both cones to form two unbounded curves (each called a branch of the hyperbola)
- #7: If the plane intersects the double right circular cones, passing through the vertex, degenerate cones are formed. When perpendicular to the base, such as that of a circle, a point is formed. When the plane is tilted, such as the case of a parabola, a line is the intersection. A hyperbola becomes two intersecting lines.