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Transcript of Conic Sections
1. What would the equation of a circle with a center at the origin look like?
2. What would the equation of the unit circle look like? (x-h) + (y-k) = r 2 2 2 Circles General form: x + y + ax + by + c = 0 2 2 How can we write this in standard form? By completing the square! Let's practice... x + y + 4x - 6y + 12 = 0 Step one: Group the x terms, group the y terms, and move the constant. (x + 4x) +(y - 6y) = -12 Step two: Complete the square for each set of parentheses. (x + 4x+ ) + (y - 6y+ ) = -12 + + 2 2 *Remember what we add to the left we must add to the right! (x + 2) + (y - 3) = 1 Step three: Factor! 2 2 What is the radius? What is the center? Let's do this one together: 2x + 2y - 8x + 12y + 9 = 0 2 2 Do this one with your partner: 2x + 2y - 16x + 8y + 10 = 0 2 2 Do this one on your own: x + y - 2x - 4y - 4 = 0 2 2 Write in standard form; find center and radius. 2 2 2 2 Ellipse The set of all points in the plane, the sum of whose distances from two fixed points is a constant. Vocabulary Foci: Fixed Points Focal Axis: Line through the foci (line) Major axis: chord lying on the focal axis (length 2a) Vertices: Points of intersection of the ellipse and the major axis Minor axis: chord through the center perpendicular to the focal axis (length 2b) Semimajor axis: half the length of the major axis (a) Semiminor axis: half the length of the minor axis (b) Ellipse Chart Write an equation for the ellipse that satisfies each set of conditions: 1. Endpoints at (-2, -2) and (-2, 8) Foci located at (-2,6) and (-2,0) 2. Endpoints (-8,4) and (4,4) foci at (-3,4) and (-1,4) 3. Endpoints of major axis at (3,2) and (3,-14) endpoints of minor axis at (-1,-6) and (7,-6) Hyperbolas Definition: the set of all points in the plane, the difference of whose distances from two fixed points is a constant. Quick talk with your group: What is the difference between the definition of a hyperbola and an ellipse? Hyperbolas Find 3 differences between hyperbolas with a horizontal transverse axis and hyperbolas with a vertical transverse axis. Steps for graphing hyperbolas: 1. Write equation in standard form. 2. Identify if it opens up and down or left and right. 3. Graph the center 4. Draw a rectangle with dimensions 2a and 2b and center ( h, k).
If hyperbola opens left and right then a tells you left and right for the rectangle, b tells you up and down.
If hyperbola opens up and down then a tells you up and down for the rectangle, b tells you left and right. 5. The diagonals of the rectangle are the asymptotes. Example for your notes: Find the coordinates of the vertices and foci, the equations of the asymptotes. Graph the hyperbola. (y - 3) - ( x + 2 ) = 1 9 ________ 2 2 Hyperbola Parabola Identify the coordinates of the vertex and focus, the equations of the axis of symmetry and directrix and the direction of opening. y = 2x - 12 x - 25 2 Write an equation given:
Vertex (3 , 1) and Focus (3, 5) Your turn: y + 6y + 12x - 15 = 0 2