1. CONIC SECTIONS Prepared by: Jessa Mae B. Edulan-Prado, MSc

2. CONIC SECTIONS A conic section (or simply conic), is a curve obtained as the intersection of the surface of a cone with a plane.

3. ACTIVITY Determine the type of conic sections that can be generated in the activity. You will start imagining trying to connect the dots of the two ice cream cones in which you can form a double-napped cone.

4. ACTIVITY Determine the type of conic sections that can be generated in the activity. You will start imagining trying to connect the dots of the two ice cream cones in which you can form a double-napped cone. 1. If you cut the double-napped cone using a plane figure horizontally, which of the following types of conic sections will be formed?

5. ACTIVITY Determine the type of conic sections that can be generated in the activity. You will start imagining trying to connect the dots of the two ice cream cones in which you can form a double-napped cone. 1.If you cut the double-napped cone using a plane figure horizontally, which of the following types of conic sections will be formed? 2.When the (tilted) plane intersects only one cone to form a bounded curve, which of the following types of conic sections will be formed?

6. ACTIVITY Determine the type of conic sections that can be generated in the activity. You will start imagining trying to connect the dots of the two ice cream cones in which you can form a double-napped cone. 1.If you cut the double-napped cone using a plane figure horizontally, which of the following types of conic sections will be formed? 2.When the (tilted) plane intersects only one cone to form a bounded curve, which of the following types of conic sections will be formed? 3.When the plane intersects only one cone to form an unbounded curve, which of the following types of conic sections will be formed?

7. ACTIVITY Determine the type of conic sections that can be generated in the activity. You will start imagining trying to connect the dots of the two ice cream cones in which you can form a double-napped cone. 1.If you cut the double-napped cone using a plane figure horizontally, which of the following types of conic sections will be formed? 2.When the (tilted) plane intersects only one cone to form a bounded curve, which of the following types of conic sections will be formed? 3.When the plane intersects only one cone to form an unbounded curve, which of the following types of conic sections will be formed? 4.When the plane (not necessarily vertical) intersects both cones to form two unbounded curves, which of the following types of conic sections will be formed?

8. Circle (Figure 1.1) - when the plane is horizontal Ellipse (Figure 1.1) - when the (tilted) plane intersects only one cone to form a bounded curve Parabola (Figure 1.2) - when the plane intersects only one cone to form an unbounded curve Hyperbola (Figure 1.3) - when the plane (not necessarily vertical) intersects both cones to form two unbounded curves (each called a branch of the hyperbola)

9. Degenerate Conics

10. CIRCLE

12. 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.

13. The term radius is both used to refer to a segment from the center C to a point P on the circle, and the length of this segment. See Figure 1.8, where a circle is drawn. It has center C(h, k) and radius r > 0. A point P(x, y) is on the circle if and only if PC = r. For any such point then, its coordinates should satisfy the following. This is the standard equation of the circle with center C(h, k) and radius r. If the center is the origin, then h = 0 and k = 0. The standard equation is then x² + y²= r².

14. Examples

15. Equation of a circle in standard form An equation of a circle whose center is at (h, k) and radius is r > 0 is: (x – h)² + (y – k)² = r². It implies that when the radius is less than or equal to 0, the equation is not a circle.

17. General Form of the Equation of the Circle If the equation of a circle is given in the general form Ax² + Ay² + Cx + Dy + E = 0, A≠0, or x² + y² + Cx + Dy + E = 0, we can determine the standard form by completing the square in both variables.

18. Examples

20. SITUATIONAL PROBLEMS INVOLVING CIRCLES 1. A street with two lanes, each 10 ft wide, goes through a semicircular tunnel with radius 12 ft. How high is the tunnel at the edge of each lane? Round off to 2 decimal places.

21. SITUATIONAL PROBLEMS INVOLVING CIRCLES 2. A seismological station is located at (0, -3), 3 km away from a straight shoreline where the x-axis runs through. The epicenter of an earthquake was determined to be 6 km away from the station. (a) Find the equation of the curve that contains the possible location of the epicenter. (b) If furthermore, the epicenter was determined to be 2 km away from the shore, find its possible coordinates (rounded off to two decimal places).

22. SITUATIONAL PROBLEMS INVOLVING CIRCLES 3. A Ferris wheel is elevated 1 m above ground. When a car reaches the highest point on the Ferris wheel, its altitude from ground level is 31 m. How far away from the center, horizontally, is the car when it is at an altitude of 25 m?

23. SITUATIONAL PROBLEMS INVOLVING CIRCLES 4. In 1986, a nuclear reactor exploded at a power plant about 110 km. north and 15 km. west of Kiev. At first, officials evacuated people within 30 km. of the power plant. Write an equation to represent the boundary of the evacuated region if the origin of the coordinated system is at Kiev.

24. SITUATIONAL PROBLEMS INVOLVING CIRCLES 5. An earthquake will be felt up to 81 miles from its epicenter. As a STEM student, you know you are located 60 miles west and 45 miles south of the epicenter. You then print a map with gridlines and locate the epicenter as (0,0). Now, a. Find the standard equation that describes the outer boundary of the earthquake. b. Would you have felt the earthquake? c. Verify your answer to part b by getting the coordinates of your location. (Use the information you have, that is 60 miles west and 45 miles south). d. How far are you from the outer boundary?

25. PERFORMANCE TASK 1 Do the indicated task individually. Apply the concepts you gained in this topic in doing this performance task. 1. Take a photo of any circular object inside your house. 2. Trace that photo in a rectangular cartesian plane with a 1-centimeter distance from each number. 3. Place the center of the photo in the origin of the rectangular Cartesian plane and measure the radius of the circle. 4. Think of a creative design for labeling the circular object in a rectangular cartesian plane. 5. Determine the standard form and general form of the equation of the circular object.

26. QUIZ 1 Identify the center and radius of the circle with the given equation in each item. Then write its standard equation (or the general form of the equation of the circle). Show your solution neatly in finding center and radius of the circle of each equation. Sketch its graph, and indicate the center. Standard formGeneral formCenterRadiusGraph 1. 2. 3. TEST I

27. Find the standard equation of the circle which satisfies the given conditions. Show your solutions. TEST II 4. a diameter with endpoints M (-9, 2) and N (15, 12) 5. center in the third quadrant tangent to both the x-axis and y-axis, radius 7

28. TEST III An earthquake will be felt up to 81 miles from its epicenter. As a STEM student, you know you are located 60 miles west and 45 miles south of the epicenter. You then print a map with gridlines and locate the epicenter as (0,0). Now, a. Find the standard equation that describes the outer boundary of the earthquake. b. Would you have felt the earthquake? c. Verify your answer to part b by getting the coordinates of your location. (Use the information you have, that is 60 miles west and 45 miles south). d. How far are you from the outer boundary?