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Warm Up Complete the square 1) 2) 3)

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Unit 5 Conic Sections The Circle

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History Conic sections is one of the oldest math subjects studied. The conics were discovered by a Greek mathematician named Menaechmus (c BC). Menaechmus’s intelligence was highly regarded… he tutored Alexander the Great.

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History Appollonius (c BC) wrote about conics in his series of books simply titled “Conic Sections”. Appollonious’ nickname was “the Great Geometer” He was the first to base the theory of all three conics on sections of one circular cone. He is also the one to assign the name “ellipse”, “parabola”, and “hyperbola” to three of the conic sections. Appollonius

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A conic section is a curve formed by the intersection of _________________________ a plane and a double cone.

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Circles The set of all points that are the same distance from the center. Standard Equation: With CENTER: (h, k) & RADIUS: r (square root) (h, k) r

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Example 1 -h -k r² Center: Radius: r ( k ), h

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Example 2 Center: Radius:

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Example 3 Center ? Radius ?

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Not In Standard form? Move all variables to one side (group like terms together) and the constant to the other side Complete the square on both variables to put it in standard form Factor as squares EX Center? Radius?

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Find the center and radius of the circle:

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Example

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HW: Write the Equation of the circle in S.F by completing the square

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Warm up Find the center and radius of the following circles

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Warm up Find the center and radius of the following circles

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MINI QUIZ Find the center and radius of the following circles

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TIME TO GRAPH!

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Center (x, y):Radius(r):(0, 0) To graph: 1.) plot the center coordinate 2.) go up, down, left & right r units 3.) Sketch a circle using these points as guides

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4.) Center at (5, -2) and a radius of 4 Write the equation for the given information and then graph.

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Circles Continued

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Tangent line and a Circle In order to write the equation of a line, we need a point on the line and the slope. In this case, we know the slope of the radius is. Since the tangent line is perpendicular to the radius, the slope of the tangent line must be. 1 st Rewrite the equation as 2 nd Insert your given point for x and y 3 rd Solve the new equation for y and put in y = mx + b form

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Example

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Systems of Equations Containing a Circle and a Line We solve them using graphing and substitution. Three possible solutions Two solutions- they intersect at two points One solution-they intersect at exactly one point No solution-they don’t intersect at all

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Solving Graphically Graph the circle using the center and the radius Solve the linear equation for “y” and graph the line using the slope and y-intercept Center: (0,0) r = 7 m = 1, y-int = -7

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Given a system of equations such as Solve y in terms of x and substitute into the circle equation. y = -x + 1 x 2 + (-x + 1) 2 = 9 x 2 + x 2 2x + 1 = 9 2x 2 2x 8 = 0 x 2 x 4 = 0 Using the Quadratic Formula x =, which gives x = 2.56 and x = Substituting these values into the linear equation yields y = and y = 2.56 respectively. Therefore the line intersects the circle in two points (2.56, -1.56) and (-1.56, 2.56).

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Warm up: Choose one and solve

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To check solutions by graphing on the TI-83 Calculators Solve both equations for y Press “y=“ and enter the equations in y1, y2, and y3 Press window. Your graphing window should be a x to y ratio of 3 to 2 (ex. 9 to 6 or 12 to 8) Press graph To find the solutions of where they intersect, Press “Zoom” “1”. Use the arrows to get close to an intersection. Press “enter”. Use the arrows to open your box to surround your intersection. Press “enter”. Press “trace” and use the arrows to move to the intersection point. Your solution will be at the bottom of the screen as x and y.

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System of Equations with Two Circles No SolutionOne solution Two Solutions We solve these by graphing

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Graphically Graph both circles on the same set of axes and find the intersection points

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Algebraically Multiply one equation by -1 and add the two equations (this will eliminate the x 2, and y 2 Solve the remaining equation for x or y Substitute this equation in an original equation and solve Substitute the answer into the equation from Step two to find the other coordinate

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To find solutions by graphing on the TI-83 Calculators Solve both equations for y (you will have 4 equations) Press “y=“ and enter the equations in y1, y2, y3 and y4 Press window. Your graphing window should be a x to y ratio of 3 to 2 (ex. 9 to 6 or 12 to 8) Press graph To find the solutions of where they intersect, Press “Zoom” “1”. Use the arrows to get close to an intersection. Press “enter”. Use the arrows to open your box to surround your intersection. Press “enter”. Press “trace” and use the arrows to move to the intersection point. Your solution will be at the bottom of the screen as x and y.

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Example

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H W Worksheet: Conic Sections: Circle

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