Conics

Conic Sections - Definition A conic section is a curve formed by intersecting cone with a plane There are four types of Conic sections

Conic Sections - Four Types

The Distance and Midpoint Formulas To find the distance between any two points (a, b) and (c, d), use the distance formula: Distance = (c – a) 2 + (d – b) 2 The midpoint of a line is halfway between the two endpoints of a line To find the midpoint between (a, b) and (c, d), use the midpoint formula: Midpoint = (a + c), (b + d) 2 2

Example Find the distance between (-4, 2) and (-8, 4). Then find the midpoint between the points. Distance = [(-8) – (-4)] 2 + (4 – 2) 2 Distance = (-8 + 4) 2 + (2) 2 Distance = (-4) 2 + (2) 2 Distance = Distance= 20 Midpoint = (-4) + (-8), Midpoint = -12, Midpoint = ( -6, 3)

Problems 1.Find the distance between (0, 1) and (1, 5). 2.Find the midpoint between (6, -5) and (-2, -7). 3.Find the value for x if the Distance = 53 and the endpoints are (-3, 2) and (-10, x). 4.If you are given an endpoint (3, 2) and midpoint (-1, 5), what are the coordinates of the other endpoint? 1) 17 2) (2, -6) 3) x = 0 or x = 4 4) (-5, 8)

Circles A circle is a set of points equidistant from a center point The radius is a line between the center and any point on the circle The equation of a circle is (x – h) 2 + (y – k) 2 = r 2 where the radius is r and the vertex is (h, k) Sometimes you need to complete the square twice to get the equation in this form (once for x and once for y) Radius (r) Vertex (k, h)

Examples 1.Find the center and radius of x 2 + y 2 + 4x – 12y – 9 = 0 and then graph the circle. x 2 + 4x +  + y 2 – 12y +  = 9 +  +  x 2 + 4x y 2 – 12y + 36 = (x + 2) 2 + (y – 6) 2 = 49 Radius = 7 and Center is (-2, 6)

Examples (cont.) 2.If a circle has a center (3, -2) and a point on the circle (7, 1), write the equation of the circle. Find the radius by the distance formula. Radius = (7 – 3) 2 + (1 – (-2)) 2 r = (4) 2 + (3) 2 r = r = 25 r = 5 The equation of the circle will be (x – 3) 2 + (y + 2) 2 = 25

Problems 1.Find the center and radius of x 2 + y 2 + 4y = 0. Then graph the circle. 2.If a circle has a center (0, 0) and a point on the circle (-2, -4) write the equation of the circle. 1) x 2 + (y + 2) 2 = 4 Center (0, -2) and Radius 2 2) x 2 + y 2 = 20#1

Parabolas A parabola is a set of points on a plane that are the same distance from a given point called the focus and a given line called the directrix The axis of symmetry is perpendicular to the directrix and passes through the parabola at a point called the vertex The latus rectum goes through the focus and is perpendicular to the axis of symmetry If the equation of the parabola begins with x= then the parabola is not a function (fails the vertical line test) Axis of Symmetry Focus Latus Rectum Vertex Directrix Parabola

Parabolas (cont.) Form of the equation y = a (x – h) 2 + kx = a (y – k) 2 + h Axis of Symmetry x = hy = k Vertex(h, k) Focus(h, k + 1/4a)(h + 1/4a, k) Directrixy = k – (1/4)ax = h – (1/4)a Direction of Opening Opens upward when a > 0 and downward when a < 0 Opens to the right when a > 0 and to the left when a < 0 Length of Latus Rectum 1/a units Important Information About the Parabolas

Example Write y = x 2 + 4x + 1 in the form y = a (x – h) 2 + k and name the vertex, axis of symmetry, and the direction the parabola opens. y = x 2 + 4x + 1 y = (x 2 + 4x +  ) + 1 –  y = (x 2 + 4x + 4) + 1 – 4 y = (x + 2) 2 – 3 Vertex: (-2, -3) Axis of Symmetry: x = -2 The parabola opens up because a = 1 so a > 0. You can always check your answers by graphing.

Problems 1.Graph the equation x 2 = 8y. 2.For the parabola y 2 = -16x name the vertex, focus, length of latus rectum, and direction of opening. Also, give the equations of the directrix and axis of symmetry. 3.Given the vertex (4, 1) and a point on the parabola (8, 3), find the equation of the parabola. 2) Vertex: (0,0) Focus: (-4,0) Latus rectum: 16 Direction: left Directrix: x = 4 Axis of symmetry: y = 0 3) y = (1/8)(x – 4) Graph for #1 Graph for #2

Ellipses An ellipse is the set of all points in a plane such that the sum of the distances from the foci is constant An ellipse has two axes of symmetry The axis of the longer side of the ellipse is called the major axis and the axis of the shorter side is the minor axis The focus points always lie on the major axis The intersection of the two axes is the center of the ellipse Major Axis Minor Axis Center Focus

Ellipses (cont.) Equation of the EllipseFoci Points Is the major axis horizontal or vertical? Center of the Ellipse (x – h) 2 + (y – k) 2 a 2 b 2 ( h + c, k) and (h – c, k) Horizontal(h, k) (x – h) 2 + (y – k) 2 b 2 a 2 (h, k + c) and (h, k – c) Vertical(h, k) = 1 Important Notes: In the above chart, c = a 2 – b 2 a 2 > b 2 always so a 2 is always the larger number If the a 2 is under the x term, the ellipse is horizontal, if the a 2 is under the y term the ellipse is vertical You can tell that you are looking at an ellipse because: x 2 is added to y 2 and the x 2 and y 2 are divided by different numbers (if numbers were the same, it’s a circle)

Example 1.Given an equation of an ellipse 16y 2 + 9x 2 – 96y – 90x = -225 find the coordinates of the center and foci as well as the lengths of the major and minor axis. Then draw the graph. 16 (y 2 – 6y +  ) + 9 (x 2 – 10x +  ) = (  ) + 9(  ) 16 (y 2 – 6y + 9) + 9 (x 2 – 10x + 25) = (9) + 9(25) 16 (y – 3) (x – 5) 2 = 144 (y – 3) 2 + (x – 5) = 1 Center: (5, 3) 16 > 9 so the foci are on the vertical axis c = 16 – 9 c = 7 Foci: ( 5 + 7, 3) and (5 – 7, 3) Major Axis Length = 4 (2) = 8 Minor Axis Length = 3 (2) = 6

Problems 1.For 49x y 2 = 784 find the center, the foci, and the lengths of the major and minor axes. Then draw the graph. 2.Write an equation for an ellipse with foci (4, 0) and (-4, 0). The endpoints of the minor axis are (0, 2) and (0, -2). 1)Foci: (0, - 33) (0, 33) Center: (0, 0) Length of major= 14 Length of minor= 8 2)X 2 + y = 1 #1

Hyperbolas A hyperbola is a set of all points on a plane such that the absolute value of the difference (subtraction) of the distances from a point to the two foci is constant The center is the midpoint of the segment connecting the foci The vertex is the point on the hyperbola closest to the center The asymptotes are lines the hyperbola can approach but never touch The transverse axis goes through the foci The conjugate axis is perpendicular to the transverse axis at the center point Hyperbola Conjugate Axis Transverse Axis Asymptotes Focus Center Vertex

Hyperbolas (cont.) Equation of HyperbolaCenterFoci PointsEquation of Asymptote VertexTransverse Axis (x – h) 2 _ (y – k) 2 a 2 b 2 (h, k)(h – c, k) and (h + c, k) y = +/- (b/a) x(h +a, k) and (h – a, k) Horizontal (y – k) 2 _ (x – h) 2 a 2 b 2 (h, k)(h, k – c) and (h, k + c) (c = a 2 + b 2 ) y = +/- (a/b) x(h, k + a) and (h, k – a) Vertical = 1 You must be looking at a hyperbola because the x 2 and y 2 terms are subtracted (x 2 – y 2 ) or (y 2 – x 2 )

Example Write the standard form of the equation of the hyperbola 144y 2 – 25x 2 – 576y – 150x = Then find the coordinates of the center, the vertices, the foci, and the equation of the asymptotes. Graph the hyperbola and the asymptotes. 144(y 2 – 4y +  ) – 25(x 2 + 6x +  ) = (  ) + 25(  ) 144(y 2 – 4y + 4) – 25(x 2 + 6x + 9) = (4) + 25(9) 144(y – 2) 2 – 25(x + 3) 2 = 3600 (y-2) 2 _ (x + 3) = 1 Center: (-3, 2) a = 5 so the vertices are (-3, 7) and (-3, -3) a 2 + b 2 = c = c 2 c = 13 The foci are (-3, 15) and (-3, -11).

Example (cont.) Asymptotes have the formula y = +/- a/b x and we have center (-3, 2) and slopes +/- 5/12. y – 2 = 5/12 (x + 3)y – 2 = -5/12 (x + 3) y – 2 = (5/12) x + 15/12y – 2 = (-5/12) x + -15/12 y = (5/12) x + 13/4y = (-5/12) x + 3/4

Problems Find the coordinates of the vertices and the foci. Give the asymptote slopes for each hyperbola. Then draw the graph. 1)x 2 _ y )25x 2 – 4y 2 = 100 = 1 1)Vertices: (-3, 0) (3, 0) Foci: (- 58, 0) ( 58, 0) Slope = +/- 7/3 2)Vertices: (-2, 0) (2, 0) Foci: ( 29, 0) (- 29, 0) Slope = +/- 5/2 1)2)

Conic Sections Circles, ellipses, parabolas, and hyperbolas are all formed when a double cone is sliced by a plane The general equation of any conic section is : Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 The standard equations for each specific conic section are listed in previous sections If B = 0 and you look at A and C in the equations: Conic Section Name Relationship of A and C ParabolaA = 0 or C= 0, but never both equal to 0 CircleA = C EllipseA and C have the same sign, but A = C HyperbolaA and C have opposite signs

Example Identify 9x y 2 – 54x + 64y + 1 = 0 as one of the four conic sections. Then graph the conic section. 9x y 2 – 54x + 64y = -1 9 (x 2 – 6x +  ) + 16(y 2 + 4y +  ) = (  ) + 16(  ) 9 (x 2 – 6x + 9) + 16(y 2 + 4y + 4) = (9) + 16(4) 9(x – 3) (y + 2) 2 = 144 (x – 3) 2 (y + 2) This conic section is an ellipse. += 1

Problems Write the equation in standard form and decide if the conic section is a parabola, a circle, an ellipse, or a hyperbola. Then graph the equation. 1)x 2 + y 2 + 6x = 7 2)5x 2 – 6y 2 – 30x – 12y + 9 = 0 1) (x + 3) 2 + (y) 2 = 16 circle 2) (x – 3) 2 _ (y + 1) 2 hyperbola 6 5 = 1 1)2)