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To each conic section (ellipse, parabola, hyperbola, circle) there is a number called the eccentricity that uniquely characterizes the shape of the curve. Conic Sections: Eccentricity

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If e = 1, the conic is a parabola. If e = 0, the conic is a circle. If e < 1, the conic is an ellipse. If e > 1, the conic is a hyperbola. Conic Sections: Eccentricity

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For both an ellipse and a hyperbola where c is the distance from the center to the focus and a is the distance from the center to a vertex. Conic Sections: Eccentricity

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Classifying Conics 10.6 What is the general 2 nd degree equation for any conic? What information can the discriminant tell you about a conic?

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The equation of any conic can be written in the form- Called a general 2 nd degree equation

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Circles Can be multiplied out to look like this….

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Ellipse Can be written like this…..

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Parabola Can be written like this…..

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Hyperbola Can be written like this…..

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How do you know which conic it is when its been multiplied out? Pay close attention to whose squared and whose not… Look at the coefficients in front of the squared terms and their signs.

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Circle Both x and y are squared And their coefficients are the same number and sign

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Ellipse Both x and y are squared Their coefficients are different but their signs remain the same.

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Parabola Either x or y is squared but not both

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Hyperbola Both x and y are squared Their coefficients are different and so are their signs.

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You Try! 1.Ellipse 2.Parabola 3.Hyperbola 4.Circle 5.Hyperbola 6.Parabola 7.Circle 8.Ellipse 9.Hyperbola 10.Ellipse

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When you want to be sure… of a conic equation, then find the type of conic using discriminate information: Ax 2 +Bxy +Cy 2 +Dx +Ey +F = 0 B 2 4AC < 0, B = 0 & A = CCircle B 2 4AC < 0 & either B0 or ACEllipse B 2 4AC = 0Parabola B 2 4AC > 0Hyperbola

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Classify the Conic 2x 2 + y 2 4x 4 = 0 Ax 2 +Bxy +Cy 2 +Dx +Ey +F = 0 A = 2 B = 0 C = 1 B 2 4AC = 0 2 4(2)(1) = 8 B 2 4AC < 0, the conic is an ellipse

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Graph the Conic 2x 2 + y 2 4x 4 = 0 2x 2 4x + y 2 = 4 2(x 2 2x +___)+ y 2 = 4 + ___(2/2) 2 = 1 2(x 2 2x +1)+ y 2 = 4 + 2(1) 2(x1) 2 + y 2 = 6 V(1±6), CV(1±3) Complete the Square

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Steps to Complete the Square 1. Group xs and ys. (Boys with the boys and girls with the girls) Send constant numbers to the other side of the equal sign. 2. The coefficient of the x 2 and y 2 must be 1. If not, factor out. 3. Take the number before the x, divide by 2 and square. Do the same with the number before y. 4. Add these numbers to both sides of the equation. *(Multiply it by the common factor in #2) 5. Factor

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Write the equation in standard form by completing the square

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What is the general 2 nd degree equation for any conic? What information can the discriminant tell you about a conic? B 2 - 4AC < 0, B = 0, A = CCircle B 2 - 4AC < 0, B 0, A CEllipse B 2 - 4AC = 0,Parabola B 2 - 4AC > 0Hyperbola

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Assignment 10.6 Page 628, odd

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