Warm-Up. Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) 2) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,2)

Transcription

1 Warm-Up Write the standard equation of the circle with the given radius and center. 1) 9; (0,0) ) 1; (0,5) 3) 4; (-8,-1) 4) 5; (4,)

2 8.4 Graph and Write Equations of Ellipses What are the major parts of an ellipse? What are the standard forms of an ellipse equation? How do you find the value of the foci c when given a & b? How do you graph an ellipse? How do you find the equation of an ellipse from a graph?

3 Definition of Ellipse An ellipse is the set of all points P in a plane such that the sum of the distances from P to two fixed points, F1 and F, called the foci, is a constant. P F1 F F1P + FP = a

4 Standard Equation of an Ellipse Horizontal Major Axis: x + a y y b =1 V1( a, 0) (0, b) a > b a b = c V (a, 0) x O F1( c, 0) (0, b) F (c, 0) length of major axis: a length of minor axis: b a = vertices b= co-vertices c= foci

5 Standard Equation of an Ellipse Vertical Major Axis: x + b y a y =1 a > b a b = c V (0, a) F (0, c) ( b, 0) O (b, 0) x length of major axis: a length of minor axis: b F1(0, c) V1(0, a)

6 Graph the equation 4x + 5y = 100. Identify the vertices, co-vertices, and foci of the ellipse. SOLUTION STEP 1 Rewrite the equation in standard form. 4x + 5y = 100 Write original equation. 4x 5x = y x = 1 Divide each side by 100. Simplify.

7 y x = 1 STEP Identify the vertices, co-vertices, and foci. Note that a = 5 and b = 4, so a = 5 and b =. The denominator of the x - term is greater than that of the y - term, so the major axis is horizontal. The vertices of the ellipse are at (+a, 0) = (+5, 0). The co-vertices are at (0, +b) = (0, +). Find the foci. C = a b = 5 = 1, so c = 1 The foci are at ( + 1, 0), or about ( + 4.6, 0). Draw the ellipse that passes through each vertex and co-vertex.

8 Graph the equation. Identify the vertices, co-vertices, and foci of the ellipse. 1. y x = 1 SOLUTION y x STEP 1 The equation is in standard form = 1 STEP Equations. Major Axis Vertices Co - vertices y x 0, + 3 Horizontal + 4, 0 + = The vertices of the ellipse are at (+ 4, 0) and co-vertices are at (0, + 3). Find the foci. c = a b = 4 3 = 7, so c = 7 The foci are at ( + 7, 0).

9 . y x = 1 SOLUTION y x STEP 1 The equation is in standard form = 1 Major Axis Vertices Co - vertices STEP Equations. y x Vertical 0, , = 1 The vertices of the ellipse are at (0, + 7) and co-vertices are at (+ 6, 0). Find the foci. c = a b = y 6 = 13, so c = 13 The foci are at (0, + 13 ). Draw the ellipse that passes through each vertex and co-vertex.

10 3. 5x + 9y = 5 SOLUTION STEP 1 Rewrite the equation in standard form. 5x + 9y 5x 5 + x 9 + = 5 Write original equation. 9y = 1 5 y 5 = 1 Divide each side by 5. Simplify. STEP Equations. x 3 y + 5 Major Axis = 1 Vertical The vertices of the ellipse are at (0 + 5), and co-vertices are at (+ 3, 0). Find the foci. c = a b = 5 9 = 16 so c = +4 The foci are at (0, + 4). Vertices (0 + 5), Co - vertices + 3, 0

11 Example Write the standard equation for an ellipse with foci at (-8,0) and (8,0) and with a major axis of 0. Sketch the graph. length of major axis: a a = 0, so a = a b = c 10 b = 8 b = b = 36, so b = 6 x y + =

12 Write an equation of the ellipse that has a vertex at (0, 4), a co-vertex at ( 3, 0), and center at (0, 0). SOLUTION Sketch the ellipse as a check for your final equation. By symmetry, the ellipse must also have a vertex at (0, 4) and a co-vertex at (3, 0). Because the vertex is on the y - axis and the co-vertex is on the x - axis, the major axis is vertical with a = 4, and the minor axis is horizontal with b = 3. y x An equation is = x y = 1 or

13 Write an equation of the ellipse that has a vertex at ( 8, 0), a focus at (4, 0), and center at (0, 0). SOLUTION Make a sketch of the ellipse. Because the given vertex and focus lie on the x - axis, the major axis is horizontal, with a = 8 and c = 4. To find b, use the equation c = a b. 4 = 8 b b = 8 4 = 48 b = 48, or 4 3 ANSWER An equation is x + 8 (4 y x y = 1 or = 1 3)

14 Example Find the vertices and co-vertices of the ellipse. x y + = vertices: (0,7) and (0,-7) co-vertices: (4,0) and (-4,0)

15 Example Write the standard equation of the ellipse. length of major axis: a a = 16, so a = 8 length of minor axis: b b = 8, so b = 4 x y + =

16 What are the major parts of an ellipse? Vertices, co-vertices, foci, center, major & minor axis What are the standard forms of an ellipse equation? How do you find the value of the foci c when given a & b? c= a b How do you graph an ellipse? Plot the vertices, the co-vertices and draw the ellipse. How do you find the equation of an ellipse from a graph? a = vertex of major axis goes into the denominator under major axis letter, and b = co-vertex of minor axis goes into the denominator under the minor axis letter in the formula.

17 HW 8.4 pg 513, #3-13 odd, 17-3 all pg 531 #7, 10, 17, 18

18 Warm-Up Write the standard equation for an ellipse with foci at (-5,0) and (5,0) and with a major axis of 18. Sketch the graph.

19 8.4 Day Ellipses What is the standard equation for an ellipse if the vertex has been translated?

20 Standard Equation of a Translated Ellipse Horizontal Major Axis: (x h) (y k) + =1 a b a > b a b = c length of major axis: a length of minor axis: b

21 Standard Equation of a Translated Ellipse Vertical Major Axis: (x h) (y k) + =1 b a a > b a b = c length of major axis: a length of minor axis: b

22 SOLUTION (x ) (y 1) Graph + = STEP 1 Find h, k, a, and b. The center is at (h, k) = (, 1). Because a = 16 and b = 9, you know that a = 4 and b = 3. STEP Plot the center, vertices and co-vertices. Vertices are at (h + a,k) are (6,1) and (,1)and co-vertices are at (h,k +b) are (,4) and (, ) STEP 3 Draw the ellipse that panes through each vertices and co-vertices

23 Write an equation of the ellipse with foci at (1, ) and (7, ) and co-vertices at (4, 0) and (4, 4). SOLUTION STEP 1 Determine the form of the equation. First sketch the ellipse. The foci lie on the major axis, so the axis is horizontal. The equation has this form: (x h) (y k) + = 1 a b STEP Identify h and k by finding the center, which is halfway between the foci (or the co-vertices) (h, k) =( 1 + 7, + ) = (4, ) STEP 3 Find b, the distance between a co-vertex and the center (4, ), and c, the distance between a focus and the center. Choose the co-vertex (4, 4) and the focus (1, ): b = 4 = and c = 1 4 = 3.

24 STEP 4 Find a. For an ellipse, a = b + c = + 3 = 13, so a = 13 ANSWER The standard form of the equation is (x 4) (y ) + =

25 Example An ellipse is defined by the equation 4x + 9y 16x + 18y = 11. Write the standard equation and identify the coordinates of the center, vertices, co-vertices, and foci. Sketch the graph of the ellipse. 4x 16x + 9y + 18y = 11 4(x 4x) + 9(y + y) = 11 4(x 4x + 4) + 9(y + y + 1) = (4) + 9(1) 4(x ) + 9(y + 1) = 36

26 Identify the coordinates of the center, vertices, co-vertices, and foci. Sketch the graph of the ellipse. center (h,k) = (, -1) a = 9 -> a = 3, b = 4 -> b = a b = c vertices (h±a, k) co-vertices (h,k±b) 9-4 = c (±3, -1) (,-1±) 5 = c (5,-1) and (-1,-1) (,1) and (,-3) 5 = c foci (h±c, k) (± 5, -1) (4.4,-1) and (-.4,-1)

27 Write an equation of the ellipse with foci at (3,5) and (3, 1) and vertices at (3,6) and (3, ). Plot the given points and make a rough sketch. The ellipse has a vertical axis, so its equation is in the form of: 6 Find the center which is halfway between the vertices Find a and c a = 6 = c = 5 =3 Find b using b= a c 4 6

28 What is the standard equation for an ellipse if the vertex has been translated?

29 HW 8.4 pg 513, #3-13 odd, 17-3 all pg 531 #7, 10, 17, 18