10.2 Ellipses.

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Presentation transcript:

10.2 Ellipses

Ellipse – a set of points P in a plane such that the sum of the distances from P to 2 fixed points (F1 and F2) is a given constant K. Fixed points - foci

Major axis – the segment that contains the foci and has its endpoints on the ellipse. Endpoints of major axis are vertices Midpoint of major axis is the center of the ellipse. Minor axis – perpendicular to major axis at the center Endpoints of minor axis are co-vertices

Vertices: Co-Vertices:

a > b Write on your note card! Length from center to foci = c c2 = a2 – b2 Foci are always on major axis

Write an equation if a vertex is (0, -4) and a co-vertex is (3, 0) and the center is (0, 0)

Write an equation if centered at the origin and is 20 units wide and 10 units high.

Remember… Equation of a circle: Center (h,k)

Write this down on your note card. Remember, a > b

Eccentricity of an Ellipse Measures how ‘circular’ the ellipse is (describes the shape of the ellipse.) If e is close to 0 then foci are near center and more round. If e is close to 1 then foci are far from center and ellipse is elongated. Write on your note card!

Find center, foci, length of major and minor, vertices and co-vertices and graph.

Find center, foci, length of major and minor, vertices and co-vertices and graph. x2 + 4y2 – 6x – 16y – 11 = 0

Find center, foci, length of major and minor, vertices and co-vertices Find center, foci, length of major and minor, vertices and co-vertices. 25x2 + 4y2 - 150x + 40y + 225 = 0

Write an equation of the ellipse with the given characteristics: Center at (-4, 1), vertical major axis 18 units long, minor axis 12 units long

Write an equation of the ellipse with the given characteristics: Foci at (-1, 0) and (1, 0) and a = 4

Write an equation of the ellipse with the given characteristics: Foci at (3, 5) and (1, 5) and eccentricity 1/4

Write an equation of the ellipse with the given characteristics: Tangent to the x and y-axes and has center at (4, -7)

What happens if the denominators are equal???? IT’S A CIRCLE!!!!!!

Pg 781 #1-4,15,16,19,23,24