Lesson 1 - Circles and its Real Life Applications

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1. REVIEW: Circle is already introduced in Grade 10 Mathematics. In the accompanying figure of a circle, draw the following:

2. EXPLORE: A seismological station is located at (0, −3), 3 km away from a straight shore line where the x-axis runs through. The epicenter of an earthquake was determined to be 6 km away from the station. Illustrate the location of the seismological station using the coordinate plane. Determine the radius of the curve that contains the possible location of the epicenter. Find the standard equation of the curve that contains the possible location of the epicenter. If furthermore, the epicenter was determined to be 2 km away from the shore, find its possible coordinates (rounded o↵ to two decimal places).

3. CIRCLES Lesson 2:

4. LEARNING COMPETENCIES: Define a circle. D etermine the standard form of equation of a circle .

5. DISCUSSION: A circle is a set of all points in a plane that are equidistant from a given point C in the plane.

6. DISCUSSION:

7. DISCUSSION: STANDARD EQUATION OF A CIRCLE The standard equation of the circle with center C (h , k ) and radius, r is… (x – h) 2 + (y – k) 2 = r 2 . If the center of the circle is on the origin, then h = 0 and k = 0 . The standard equation is then x 2 + y 2 = r 2

8. SAMPLE PROBLEMS: In each item, determine the standard equation of the circle satisfying the given conditions. Center at the origin, radius of 4. Center (−4, 3), radius . Circle A in figure 4 . Circle B in figure 4. Center (-1,-2) and passes through P( 5, 2 ).  

9. DISCUSSION: Tips: A circle with center C (h, k) and tangent to… x- axis, the radius, r = /k/. y-axis, the radius, r = /h/. line y = b, r = /b – k/. line x = a, r = /a – h/.

10. DISCUSSION: Tips: A circle with center (h, k) and tangent to a line, Ax + By + C = 0 , the radius r, can be computed using the distance formula between a point and a line:

11. DISCUSSION: Note: The coordinates of the midpoint ( X m , Y m ) between two points (X 1 , Y 1 ) and (X 2 , Y 2 )can be determined using the formula:

12. SAMPLE PROBLEMS: In each item, determine the standard equation of the circle satisfying the given conditions. Center (5,−6), tangent to y-axis . Center (5,−6), tangent to the x-axis. Center (5,−6), tangent to line y = -3. Center (5,−6), tangent to line x = -7. It has a diameter with endpoints a(−1, 4) and b(4, 2).

13. INTEGRATION: Solve the following real life problems. A city planner is designing a circular park with a fountain located at the center of the park. The fountain is at the coordinates (2,3 ) on a coordinate grid that represents the park's layout. If the park's radius is 50 meters, find the equation of the circle representing the park's boundary .

14. INTEGRATION: Solve the following real life problems. 2. A satellite dish is installed at a point (−5,−1 ) on a flat rooftop. The dish covers a circular area with a diameter of 200 meters. Determine the equation of the circle that describes the coverage area of the satellite dish.

15. INTEGRATION: Solve the following real life problems. 3. A street with two lanes, each 10 ft. wide, goes through a semicircular tunnel with a radius 12 ft. How high is the tunnel at the edge of each lane?

16. DISCUSSION: Expanding the standard equation of circle will give its general equation. The general equation of a circle is expressed as Ax 2 + Ay 2 + Cx + Dy + E = 0.

17. SAMPLE PROBLEMS: Solve the following problems. Determine the standard equation of the given general equation of the circle. 16x 2 + 16y 2 + 96x − 40y – 315 = 0. The equation of the circle is expressed x 2 + y 2 – 5x + 4y = 46. Reduced its equation into its standard form and sketch its graph.

18. SAMPLE PROBLEMS: Solve the following problems. Circle A is concentric with Circle B having an equation of x 2 + y 2 – 8x – 10y + 16 = 0. Its area is 4 times the area of Circle B. Find the standard equation of circle A. Identify the center and radius of the given equation of the circle and sketch its graph. x 2 + y 2 + 8y = 33 4x 2 + 4y 2 + 16x + 40y + 67 =

19. ACTIVITY NO. 2: In each item, determine the standard equation of the circle satisfying the given conditions. Center at the origin, contains (0, 3). Center (-2, -3), tangent to x-axis. Contains the points (-2, 0) and (8, 0) with diameter 10 cm. Identify the center and radius of the circle using its equation 4x 2 + 12x + 4y 2 + 16y – 11 = 0 and sketch its graph .