*EllipsesPre Calculus

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Write equations of ellipses in standard form and graph ellipses.

Given equations of ellipses, find key features.

Find eccentricities of ellipses.

What You Should Learn

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*Plan for todayReview of Homework Quick ReviewLittle Quiz LinesEllipses HomeworkPage 722 # 1-6 all (matching), 7, 11,15, 21, 39 47 odd

*So FarDouble napped coneIntersection of plane and the cone created 4 basic sections

*LinesInclination If a non horizontal line has an inclination of and slope m, then tan = m. Remember if the slope, m, is negative, find and add 180o

Angle between two lines If two non perpendicular lines have m1 and m2, the angle between the two lines is:

Distance between the point (x1, y1) and a line Ax + By + C = 0 is:

*Recognizing a ConicAx2 + Cy2 + Dx + Ey + F = 0

AC = 0 A or C is zero - no x2 or no y2 termParabolaA = C A is equal to C, same value CircleAC > 0 A and C have the same sign but have different valuesEllipseAC < 0 A and C have different signs Hyperbola

*CirclesStandard form of a circle with the center at (0, 0) and r is the radius: x2 + y2 = r2 Standard form of a circle with the center at (h, k) and radius of r:(x h)2 + (y k)2 = r2

*ParabolasOrientationVertex (h, k) DirectrixFocusAxis of symmetrypEquations:

*You should be able toIdentify key features of linesRecognize a ConicGiven an equation of a CircleFind the center and radiusGiven key information about a circle, find the equation in standard formGiven an equation of a ParabolaFind the orientation, focus, vertex, directrixSketchGiven key information about a parabola, find the equation in standard form

Quiz Time

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*An ellipse is the set of all points in the plane for which the sum of the distances to two fixed points (called foci) is a positive constant. The major axis is the line segment passing through the foci with endpoints (called vertices) on the ellipse. The minor axis is the line segment perpendicular to the major axis passing through the center of the ellipse with endpoints on the ellipse. The midpoint of the major axis is the center of the ellipse.The sum of the distance from any point on the ellipse to each of the foci remains constant.

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*Key Identifiers of an Ellipsea is the distance from the center to the end of the major axis. Therefore the entire length of the major axis is 2a. The center is the midpoint of the major axis. a is always the largest number.

b is the distance from the center to the end of the minor axis. Therefore the entire length of the minor axis is 2b. The center is the midpoint of the minor axis.

c is the distance from the center to one of the two focus points. The center is the midpoint of the two focus points (foci).

c2 = a2 b2

*The standard form for the equation of an ellipse with center atthe origin and a major axis that is horizontal is: , with:

vertices: (a, 0), (a, 0) andfoci: (c, 0), (c, 0)where c2 = a2 b2

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*The standard form for the equation of an ellipse with center at the origin and a major axis that is vertical is: , with:

vertices: (0, a), (0, a) andfoci: (0, c), (0, c)where c2 = a2 b2

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*Example: Sketch the ellipse with equation 25x2 + 16y2 = 400 and find the vertices and foci. 1. Put the equation into standard form. divide by 4002. Since the denominator of the y2-term is larger, the major axis is vertical. 3. Vertices: (0, 5), (0, 5) 4. The minor axis is horizontal and intersects the ellipse at (4, 0) and (4, 0). 5. Foci: c2 = a2 b2 (5)2 (4)2 = 9 c = 3 foci: (0, 3), (0,3)

So, a = 5 and b = 4.

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*The standard form for the equation of an ellipse with center atthe (h, k) and a major axis that is horizontal is:

ba(h, k)

vertices: (h a, k), (h + a, k) andfoci: (h c, k), (h + c, k)where c2 = a2 b2

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a(h, k)c(h b, k)The standard form for the equation of an ellipse with center atthe (h, k) and a major axis that is vertical is: vertices: (h , k a), (h, k + a) andfoci: (h, k c), (h, k + c)where c2 = a2 b2

(h + b, k)

bba

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*Note on OrientationTo recognize the difference between a vertically oriented and horizontally oriented ellipse:Put the equation is standard formIf the largest number is under the x2 term it is horizontalIf the largest number is under the y2 term it is verticalRemember a is always the largest

*Example: Find the standard form of the equation of the ellipse having foci at (0,1), and (4, 1) and a major axis length of 6. 1. Identify what information you have. The length of the major axis = 2a so a =33. Choose the equation based upon orientation of the ellipse, since a is larger, the major axis is horizontal. Or, since the foci are on a horizontal line, it is horizontal.2. Calculate b: a2 b2 = c24. Put information into the equation. Since the foci occur at (0, 1) and (4, 1) the center of the of the ellipse is (2, 1) - midpoint of the fociThe distance from the center to a focus point is 2, c = 2a2 c2 = b2 32 22 = b2 5 = b2

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*Example: Sketch a graph of the ellipse: x2 + 4y2 + 6x 8y + 9 = 0Put the equation into standard form by completing the square.

2. Since the denominator of the x2-term is larger, the major axis is horizontal.

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* Example: Sketch a graph of the ellipse: x2 + 4y2 + 6x 8y + 9 = 03. To graph this we need to know key information:Since a2 = 4, a = 2, the major axis vertices are: (h + 2, k) and (h 2, k) (-1, 1) and ( -5, 1)4. The foci can be found but are not needed to graph: c2 = a2 b2 (2)2 (1)2 = 3 HorizontalThe center is (-3,1)Since b2 = 1, b = 1, the minor axis vertices are: (h , k + 1) and (h , k + 1) (-3, 2) and ( -3, 0)

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*(-3, 1)

The major axis vertices are: (-1, 1) and ( -5, 1)The minor axis vertices are:(-3, 2) and ( -3, 0)The center is (-3,1)

*Example: Find the center, vertices and foci of the ellipse: 4x2 + y2 + 8x +4y 8 = 0Put the equation into standard form by completing the square.

2. Since the denominator of the y2-term is larger, the major axis is vertical.

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* Example: Find the center, vertices and foci of the ellipse: 4x2 + y2 + 8x +4y 8 = 03. To graph this we need to know key information:Since a2 = 16, a = 4, the major axis vertices are: (h , k + 4) and (h , k 4) (1, 2) and ( 1, -6)4. The foci can be found but are not needed to graph: c2 = a2 b2 (4)2 (2)2 = 12 VerticalThe center is (1,-2)Since b2 = 4, b = 2, the minor axis vertices are: (h +2 , k) and (h 2, k) (3, -2) and ( -1, -2)

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*EccentricityThe eccentricity e of an ellipse is given by the ratio:Eccentricity is used to measure the ovalness of an ellipse.0 < e < 1 for every ellipse.Because the foci of an ellipse are located along the major axis between the vertices and the center, it follows that:0 < c < aFor an ellipse that is nearly circular, the foci are close to the center and the ration c/a is small. For an elongated ellipse the foci are close to the vertices and the ratio c/a is close to 1.

*Eccentricity with Circles and ParabolasEllipses have an eccentricity between zero and one.

Circles are really ellipses with an eccentricity of zero. The two foci coincide and create the center.

Parabolas always have an eccentricity of one.

*Another NoteThe area of a circle is: r2 The area of an ellipse is: ab

*Wrap up

*EllipsesHorizontal orientationCenter (h, k)Vertex (h a, k) (major)Vertex (h, k b) (minor)Focus (h c, k)Length of major axis 2aLength of minor axis 2ba2 b2 = c2 e = c/aVertical orientationCenter (h, k)Vertex (h, k a) (major)Vertex (h b, k) (minor)Focus (h, k c)Length of major axis 2aLength of minor axis 2ba2 b2 = c2 e = c/a

*Homework 41

Page 722 # 1-6 all (matching), 7, 11,15, 21, 39 47 odd

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