INTRODUCTION TO CONIC SECTIONS AND CIRCLE.

Conic Sections

1. Precalculus: CONIC SECTIONS: CIRCLES

2. Session Objectives For this module , SHS students in Precalculus are expected to develop the following learning competencies: illustrate the different types of conic sections: parabola, ellipse, circle, hyperbola, and degenerate cases; define a circle; determine the standard form and general form of the equation of a circle; and 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:

6. What is DOUBLE NAPPED CONE? It is where the conic sections are formed. Formed the conic sections by cutting it by a plane figure in different ways.

7. What are CONIC SECTIONS ? CIRCLE ELLIPSE PARABOLA HYPERBOLA

8. DEGENERATE CASES

9. GROUP ACTIVITY! Each conic section will be assigned to each group. Each group will make their own definition of the assign conic section and where it is usually seen in everyday living. Group 1 – Circle Group 2 – Ellipse Group 3 – Parabola Group 4 – Hyperbola

10. CIRCLES A circle is the set of all points equidistant to a fixed point. radius center

11. EQUATION OF A CIRLE Consider a circle plotted on the Cartesian Coordinate plane. Determine the coordinates of the center and its radius.

12. 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?    

13. 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  

14. STANDARD FORM of the equation of a circle Give the standard form of the equation of the circle described in each item below. Center at the origin, and containing and containing  

15. EQUATION OF A CIRLE Solve for the equation of the following circles: circle A circle B center (5,−6), tangent to the y-axis center (5,−6), tangent to the x-axis has a diameter with endpoints A(−1, 4) and B(4, 2)

16. IN, ON, or OUT Consider the circle with equation, Determine the location of the following points relative to the circle. P (-2, 14) Q (-6, 6) R (-10, 9) S (7, 9)  

17. GENERAL FORM of the equation of a circle The general form of the equation of a circle is w here  

18. GENERAL FORM of the equation of a circle Transform the following equations into the general form:  

19. Transforming equations Transform the following equations into the standard form:  

20. 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 binomials