Circles and ellipses

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<ul><li> 1. Lesson 10.3-10.4 </li></ul> <p> 2. Slicing these cones with a plane at different angles produces different conic sections. 3. For example, you can describe a circle as a locus of points that are a fixed distance from a fixed point. Definition of a Circle A circle is a locus of points P in a plane, that are a constant distance, r, from a fixed point, C. Symbolically, PC r. The fixed point is called the center and the constant distance is called the radius. 4. EX 1 Write an equation of a circle with center (3, -2) and a radius of 4. 2 2 2 +x h y k r 22 2 3 + 2 4x y 2 2 3 + 2 16x y 5. EX 2 Write an equation of a circle with center (-4, 0) and a diameter of 10. 2 2 2 +x h y k r 2 2 2 4 + 0 5x y 2 2 4 +y 25x 6. EX 4 Find the coordinates of the center and the measure of the radius. 2 2 2 6 + 3 25x y 7. 1. Move the x terms together and the y terms together. 2. Move C to the other side. 3. Complete the square (as needed) for x. 4. Complete the square(as needed) for y. 5. Factor the left &amp; simplify the right. 8. 2 2 4 6 3x x y y 2 2 4 6 3 0x y x y Center: (-2, 3) radius: 4 2 2 2 3 16x y 2 2 4 6 9394 4x x y y 9. Find the equation of the circle whose endpoints of a diameter are (11, 18) and (-13, -20): Center is the midpoint of the diameter 11 13 18 20 , 1 1, 2 2 Radius uses distance formula 2 2 1 2 1 2r x x y y 2 2 r 11 1 18 1 r 505 2 2 r 13 1 20 1 r 505 22 2 1 1 05x 5y 10. Write an equation in standard form of an ellipse that has a vertex at (0, 4), a co-vertex at (3, 0), and is centered at the origin. Since (0, 4) is a vertex of the ellipse, the other vertex is at (0, 4), and the major axis is vertical. Since (3, 0) is a co-vertex, the other co-vertex is at (3, 0), and the minor axis is horizontal. So, a = 4, b = 3, a2 = 16, and b2 = 9. + = 1 Standard form for an equation of an ellipse with a vertical major axis. (x-h) 2 b2 (y-k) 2 a2 + = 1 Substitute 9 for b2 and 16 for a2. (x-0) 2 9 (y-0) 2 16 An equation of the ellipse is + = 1. x 2 9 y 2 16 11. b) Find coordinates of vertices, covertices, foci Center = (-3,2) Horizontal ellipse since the a value is under x terms Since a = 3 and b = 2 Vertices are 3 points left and right from center (-3 3, 2) Covertices are 2 points up and down (-3, 2 2) Now to find focus points Use c = a - b So c = 9 4 = 5 c = 5 and c = 5 Focus points are 5 left and right from the center F(-3 5 , 2) 1 4 )2y( 9 )3x( 22 a) GRAPH Plot Center (-3,2) a = 3 (go left and right) b = 2 (go up and down) 12. Find the foci of the ellipse with the equation 9x2 + y2 = 36. Graph the ellipse. 9x2 + y2 = 36 Since 36 &gt; 4 and 36 is with y2, the major axis is vertical, a2 = 36, and b2 = 4. + = 1 Write in standard form. x 2 4 y 2 36 c2 = a2 b2 Find c. = 36 4 Substitute 4 for a2 and 36 for b2. = 32 The major axis is vertical, so the coordinates of the foci are (0, c). The foci are: (0, 4 2 ) and (0, 4 2). c = 32 = 4 2 </p>

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