Circles. 1 Page Hannah Province Mathematics Department Southwest Tn Community College

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1 Circles 1 Page

2 To Graph a Circle; Graphing Calculator + y = 2 2 First Solve the equation for y: x 4 y = 4-x 2 2 y = ± 4 x 2 2 Graph as two separate equations y = 4 x y = 4 x 1 2 So that the circle doesn't look flattened, press ZOOM, #5 for ZSqua Now press GRAPH. 2 Page

3 Write the standard form of the equation of the circle with center (-4,1) and radius of ( x 4) + ( y 1) = ( x+ 4) + ( y 1) = 9 Standard Form 3 (-4,1) Page

4 2 2 Find the center ( x+ 3) + ( y 4) and = 9 radius of the circle whose equation is Graph the equation. Use the graph to identify the relation s domain and range. Why is it a relation and not a function? Center(-3,4); radius=3 Domain: [-6,0]; Range:[1,7] 3 (-3,4) Example - 4 Page

5 Write the standard form of the equation of the circle with center at (-2,7) and a radius of 5. Example - Find the center and radius of the circle whose equation i below. Graph the equation. Use the graph 2 to identify 2 the relation s domain and range. ( x 6) + ( y+ 5) = 49 5 Page

6 If we take the equation from the previous problem we can multiply out the factors and move all terms to one side to get the general form of the equation of the circle. 2 2 ( x 6) + ( y+ 5) = x x y y = 2 2 x y x y = 0 49 General Form General Form Example Complete the square and write the equation in standard form. Then give the center and radius of each circle. x 2 + y 2 14x + 8y + 29 = 0 6 Page

7 Example - Complete the square and write the equation in standard form. Then give the center and radius of each circle. x 2 + y 2 4x + 12y + 15 = 0 Example - Complete the square and write the equation in standard form. Then give the center and radius of each circle. x 2 + y 2 + 6x 8y = 0 7 Page

8 The Ellipse Obtaining Conic Sections by Intersecting a Plane and a Cone 8 Page

9 Definition of an Ellipse Foci Foci Foci Foci An Ellipse can be elongated in any direction. We will limit our discussions to ellipses elongated vertically and horizontally. Sum of Red line = Sum of Blue Line 9 Page

10 The line segment that joins the vertices is the major axis; the midpoint of the major axis is the center of the ellipse. The line segment whose endpoints are on the ellipse and that is perpendicular to the major axis is called the minor axis. Standard Form of the Equation of an Ellipse a is half the length of the major axis b is half the length of the minor axis b 2 = a 2 c 2 or c 2 = a 2 b 2 "±c" is the ycoordinate of the foci if the major axis is vertical "±c" is the xcoordinate of the foci if the major axis is horizontal X-Intercepts: Set y=0 10 Page

11 x 2 a 2 = 1 x 2 = a 2 x = ±a Y-Intercepts: Set x=0 y 2 b 2 = 1 y 2 = b 2 y = ±b 11 Page

12 Using the Standard Form of the Equation of an Ellipse Ellipse with the Major Axis on the Y-axis 12 Page Province Mathematics Southwest TN Community College

13 Example Graph and locate the foci x + 4y = 36 4 y x Page Province Mathematics Southwest TN Community College

14 Example Graph and locate the foci. 2 2 x y + = y x Page Province Mathematics Southwest TN Community College

15 Example Write the standard equation of the ellipse pictured. y Page Province Mathematics Southwest TN Community College

16 Translations of Ellipses Standard form of Equations of Ellipses Centered at h,k 16 Page Province Mathematics Southwest TN Community College

17 How to Use the Center to find the Endpoints of the Axes Example 2 2 ( x+ 2) ( y 1) Graph + = Where are the foci? Where are the vertices, and endpoints of the minor axis. 4 y x Page Province Mathematics Southwest TN Community College

18 Example Write the equation of an ellipse given the following information. Find the center of the ellipse, and the endpoints of the minor axis. Vertices: (-1,-3), (-1,5) Foci: (-1,-1),(-1,3) Completing the Square to find the Standard Form of the Ellipse 18 Page Province Mathematics Southwest TN Community College

19 Example Convert the equation into standard form by completing the square. Where is the center, the vertices, foci and the endpoints of the minor axis x +9y +16x 18y = Page Province Mathematics Southwest TN Community College

20 Applications Ellipses Have Many Applications Example: The Victoria and Albert restaurant in the Grand Floridian Hotel at Disney World is an example of an elliptical room. Thus if one person whispers at one focus, another person can hear what is spoken if they are seated at the other focus. If the restaurant is 40 ft long and the ceiling is 10 ft high, find the standard form of the equation of the conic section. How far from the center of the room is the focus located? 20 Page Province Mathematics Southwest TN Community College