Name: Pre-Calculus Guided Notes: Chapter 10 Conics Section Circles A circle is _ Example 1 Write an equation for the circle with center (3, ) and radius 5. To do this, we ll need the x1 y y1 distance formula: d x In general: An equation of the circle with center (h, k) and radius r (r > 0) is. If the center is the origin, then. The general form of the equation of a circle is x + y + Dx + Ey + F = 0, where D, E, and F are constants. Example Sketch the graph of x + y 8x + y + 8 = 0. Center _ Radius _ 1
Example 3 Sketch the graph of x + y 6x y 6 = 0. Center Radius Example 4 Sketch the graph of x + y = 18. Center Radius Example 5 Sketch the graph of x + y + 6y = 0. Center Radius Example 6 Write the standard form of the equation of the circle that is tangent to the x-axis and has its center at (-5, 4).
Section 3 Ellipses An ellipse is Each of the two fixed points is a (plural: ) of the ellipse and the distances from the foci to a point P on the curve are called the. Example 1 Find the equation for an ellipse with foci F1(-4, 0) and F(4, 0) and PF1 +PF = 10. 3
Foci on the x-axis (c, 0) and (-c, 0) and center at the origin Equation: Foci on the y-axis (0, c) and (0, -c) and center at the origin Equation: x-intercepts: y-intercepts: The sum of the focal radii for each of its points is the constant a (a > c) c = a b x-intercepts: y-intercepts: In each case, the ellipse is symmetric with respect to both the x-axis and the y-axis and a > b Major axis: the segment of length a cut off by the ellipse Minor axis: the segment of length b that is perpendicular to the major axis at the center Vertices: the points where the ellipse cuts it major axis Example Sketch the graph of 4x + 5y = 100. horizontal/vertical shift (h, k) vertices foci major axis length minor axis length An ellipse can have its center at a point other than the origin, and its axis need not lie on the coordinate axes. Graphs of the following equations have center (h, k) and a > b. The foci are located c units to either side of the center along the major axis. x h y k x h y k a b 1 b a 1 4
Example 3 x 3 y 4 64 100 Sketch the graph of 1. horizontal/vertical shift (h, k) vertices foci major axis length minor axis length Example 4 Sketch the graph of the ellipse with equation 4x + 9y 8x 54y + 49 = 0. horizontal/vertical shift (h, k) vertices foci major axis length minor axis length Example 5 Consider the graphed ellipse. Write the equation of the ellipse in standard form and find the coordinates of the foci. 5
Example 6 Determine an equation for an ellipse on the coordinate axes with major axis of length 10 and foci at (3, 0), and (-3, 0). Section 4 Hyperbolas A hyperbola is Each fixed point is called a and the distances from the foci to a point P on the curve are called. Example 1 Write an equation for the hyperbola with foci F1(-5, 0) and F(5, 0) and with focal radii differing by 8. 6
Graphs of the following equations have center (h, k). The foci are located c units to either side of the center along the transverse axis. c = a + b x h y k y k x h a b 1 a b 1 Example Sketch the graph of 9x 5y = 5. horizontal/vertical shift (h, k) vertices foci transverse axis length conjugate axis length asymptote equations Example 3 y 1 x 5 9 5 Sketch the graph of 1. horizontal/vertical shift (h, k) vertices foci transverse axis length conjugate axis length asymptote equations 7
Example 4 Graph 4x y + 4x + 4y + 8 = 0. horizontal/vertical shift (h, k) vertices foci transverse axis length conjugate axis length asymptote equations Example 5 Find the equation of the hyperbola with foci at (1, -5) and (1, 1) and whose transverse axis is 4 units long. Rectangular Hyperbola: xy = c Example 6 Graph xy = 36. 8
Section 5 Parabolas A parabola is The fixed line is called the and the fixed point is called the _. Example 1 Write an equation for the parabola with focus F(0, 4) and directrix the line L with equation y = -. Example Write and equation for the parabola with focus F(3, ) and directrix the line L with equation x = -1. 9
Notice that the vertex of a parabola is Parabola w/ vertex (h, k) and directrix y = k p (p is the distance between the focus and the vertex) 1 x y k h 4 p x = a(y k) + h Parabola w/ vertex (h, k) and directrix x = h p (p is the distance between the focus and the vertex) 1 y x h k 4 p y = a(x h) + k Example 3 Graph x + 1 = 4y y horizontal/vertical P vertex (h, k) focus directrix equation Example 4 Graph x 8x y + 18 = 0 horizontal/vertical P vertex (h, k) focus directrix equation 10
Example 5 Write an equation for the parabola with a focus at (-1, 7), the length from the focus to the vertex is units, and has a minimum. Section 6 Rectangular and Parametric Forms of Conic Sections The equation of a conic section can be written in the form: Ax + Bxy + Cy + Dx + Ey + F = 0, where A, B, and C are not all zero. In general, the graph of Ax + Cy + Dx + Ey + F = 0 is a: when Circle Parabola Ellipse Hyperbola Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. 1. 4x 9x + y 5 = 0. 4x y + 8x 6y + 4 = 0 3. x + 4y 4x + 1y = 0 4. x + y 8x + 1y + = 0 11
So far we have discussed equations of conic sections in their rectangular form. Some conic sections can also be described parametrically. Example 1 x t 1 Graph the curve defined by the parametric equations y 4 t identify the curve by finding the corresponding rectangular equations. t x y - -1 0 1, where t. Then Example x cos t Find the rectangular equation of the curve whose parametric equations are y sin t o o where 0 t 180. Then graph the equation using arrows to indicate how the graph is traced. Example 3 Find parametric equations for the equation x y 1 9 4. 1
Section 7 Transformations of Conics Remember from earlier in this course that Th,k refers to a translation of h units horizontally and k units vertically. Example 1 Given x + 3xy 4y 5x = 0. Write the equation following a translation of T3, 1 in general form. Another type of transformation we studied this year is rotations. The figures below show an ellipse whose center is the origin and its rotation. A rotation of about the origin can be described by the matrix: If we let P(x, y) be a point on the graph of a conic section, then P (x, y ) is the image of P after a counterclockwise rotation of. The values of x and y can be found by matrix multiplication: 13
Rotation Equations To find the equation of a conic section with respect to a rotation of, replace x with and y with Example Given x + 4xy + 5y 7x y = 0. Write the equation of the graph after a rotation of =90 o. Identifying Conics by Using the Discriminant For the general equation Ax + Bxy + Cy + Dx + Ey + F = 0, if B 4AC < 0, the graph is a circle (A = C, B = 0) or an ellipse (A C or B 0). if B 4AC > 0, the graph is a hyperbola. if B 4AC = 0, the graph is a parabola. Example 3 Identify the graph of the equation 8x + 5xy 4y = -. 14
Angle of Rotation About the Origin For the general equation Ax + Bxy + Cy + Dx + Ey + F = 0, the angle of rotation about the origin can be found by if A = C, or 4 tan B, if A C A C Example 4 Identify the graph of the equation x 4xy y 6 = 0. Then find. 15