Conics can be formed by the intersection of a plane with a conical surface. If the plane passes through the Vertex of the conical surface, the intersection is a Degenerate Case (a point, a line, or two intersecting lines).

Conic Sections There are 4 types of Conics which we will investigate: Circles Parabolas Ellipses Hyperbolas

y (x, y) r (h, k) x

Graph ( ) x y + - = 1 3 16 2 by hand.

y (-1, 7) (3,3) (-5, 3) (-1,3) x (-1, -1)

Find the center, the radius and graph: x2 + y2 + 10x – 4y + 20 = 0

Find the equation of the circle with radius 3 in QI and tangent to the y-axis at ( 0 , 2 )

Find the equation of the circle with center at ( 2 , -1 ) through ( 5 , 3 )

Find the equation of the circle with endpoints of the diameter at ( 3, 5 ) and ( 3 , 1 )

Find the equation of the circle that Goes through these 3 points: (3, 4), (-1, 2), (0, 3)

A parabola is defined as the collection of all points P in the plane that are the same distance from a fixed point F as they are from a fixed line D. The point F is called the focus of the parabola, and the line D is its directrix. As a result, a parabola is the set of points P for which d(F, P) = d(P, D)

The standard form of the equation of a parabola with directrix parallel to the y-axis is (opens left or right) The standard form of the equation of a parabola with directrix parallel to the x-axis is (opens up or down) Where (h, k) represents the vertex of the parabola and “p” represents the distance from the vertex to the focus.

The Axis of Symmetry is the line through which the parabola is symmetrical. The Latus Rectum is a line segment perpendicular to the Axis of Symmetry through the focus with endpoints on the parabola. The length of the Latus Rectum is “4p”. The Latus Rectum helps define the “width” of the parabola.

focus

Parabola with Axis of Symmetry Parallel to x-Axis, Opens to the Right, a > 0. D: x = -a + h y V = (h, k) Axis of symmetryy = k F = (h + a, k) x

Parabola with Axis of Symmetry Parallel to x-Axis, Opens to the Left, a > 0. D: x = a + h y Axis of symmetry y = k F = (h - a, k) x V = (h, k)

Parabola with Axis of Symmetry Parallel to y-Axis, Opens up, a > 0. Axis of symmetry x = h y F = (h, k + a) V = (h, k) D: y = - a + k x

Parabola with Axis of Symmetry Parallel to y-Axis, Opens down, a > 0. Axis of symmetry x = h y D: y = a + k V = (h, k) F = (h, k - a) x

Find an equation of the parabola with vertex at the origin and focus (-2, 0). Graph the equation by hand and using a graphing utility. Vertex: (0, 0); Focus: (-2, 0) = (-a, 0)

The line segment joining the two points above and below the focus is called the latus rectum. Let x = -2 (the x-coordinate of the focus) The points defining the latus rectum are (-2, -4) and (-2, 4).

(-2, 4) (0, 0) (-2, -4)

Parabolas: Example Problems Write the equation of a parabola with vertex ( 0 , 0 ) and focus ( 2, 0 )

Parabolas: Example Problems Find the focus, directrix, vertex, and axis of symmetry.   y2 – 12x – 2y + 25 = 0

Parabolas: Example Problems Write the equation of the parabola with focus ( 0 , -2 ) and directrix x = 3

Parabolas: Example Problems Find the focus, directrix, vertex, and axis of symmetry.   x2 + 4x + 2y + 10 = 0

Parabolas: Example Problems Write the equation of the parabola with vertex ( 4 , 2 ) and directrix y = 5

Parabolas: Example Problems Write the equation of the parabola with directrix y = 3 and focus ( 3 , 5 )

Parabolas: Example Problems Find the focus, directrix, vertex, axis of symmetry, and length of the latus rectum.   x2 – 4x – 12y – 32 = 0  

Parabolas: Example Problems Write this equation of a parabola in standard form:

Find the vertex, focus and directrix of x y 4 8 20 0. + - =

Vertex: (h, k) = (-2, -3) p = 2 Focus: (-2, -3 + 2) = (-2, -1) Directrix: y = -2 + -3 = -5

(-6, -1) (2, -1) y = -5 (-2, -3) (-2, -1)

An ellipse is the collection of points in the plane the sum of whose distances from two fixed points, called the foci, is a constant. Minor Axis y P = (x, y) Major Axis x Focus1 Focus2

The standard form of the equation of an ellipse with major axis parallel to the x-axis is The standard form of the equation of an ellipse with major axis parallel to the y-axis is

(h,k) is the center of the ellipse For any ellipse, “2a” represents the distance along the major axis (a is always greater than b) “2b” represents the distance along the minor axis “c” represents the distance from the center to either focus (the foci of an ellipse are always along the major axis)

Ellipse with Major Axis Parallel to the x-Axis where a > b and b2 = a2 - c2. y (h + c, k) (h - c, k) Major axis (h - a, k) (h, k) (h + a, k) x

Ellipse with Major Axis Parallel to the x-Axis The ellipse is like a circle, stretched more in the “x” direction y Focus 2 Focus 1 Major axis (h, k) x

Ellipse with Major Axis Parallel to the y-Axis where a > b and b2 = a2 - c2. (h, k + a) (h, k + c) (h, k) (h, k - c) x Major axis (h, k - a)

Ellipse with Major Axis Parallel to the y-Axis The ellipse is like a circle, stretched more in the “y” direction y Focus 1 (h, k) Focus 2 x Major axis

Ellipses: Example Problems Sketch the ellipse and find the center, foci, and the length of the major and minor axes:

Ellipses: Example Problems Find the center and the foci. Sketch the graph.

Ellipses: Example Problems Write the equation of the ellipse with center ( 0 , 0 ), a horizontal major axis, a = 6 and b = 4

Ellipses: Example Problems Write the equation of the ellipse with x-intercepts  and y-intercepts

Ellipses: Example Problems Write the equation of the ellipse with foci ( -2 , 0 ) and ( 2 , 0 ), a = 7

Ellipses: Example Problems Write the equation of this ellipse

Ellipses: Example Problems Find the center, foci, and graph the ellipse: 16x2 + 4y2 – 96x + 8y + 84 = 0

Ellipses: Latus Rectum The length of the Latus Rectum for an Ellipse is By knowing the Latus Rectum, it makes the graph of the ellipse more accurate Latus Rectum Latus Rectum (h, k)

Ellipses: Latus Rectum Use the length of the latus rectum in Graphing the following ellipse:

Distance from center to focus is 5, so c = 5 Find an equation of the ellipse with center at the origin, one focus at (0, 5), and a vertex at (0, -7). Graph the equation by hand Center: (0, 0) Major axis is the y-axis, so equation is of the form Distance from center to focus is 5, so c = 5 Distance from center to vertex is 7, so a = 7

(0, 7) FOCI (0, -7)

Center: (h, k) = (-4, 2) Major axis parallel to the x-axis Vertices: (h + a, k) = (-4 + 3, 2) or (-7, 2) and (-1, 2) Foci: (h + c, k) =

(-4, 4) V(-1, 2) V(-7, 2) F(-6.2, 2) F(-1.8, 2) C (-4, 2) (-4, 0)

A hyperbola is the collection of points in the plane the difference of whose distances from two fixed points, called the foci, is a constant.

The standard form of the equation of a hyperbola with transverse axis parallel to x-axis is The standard form of the equation of a hyperbola with transverse axis parallel to y-axis is

(h,k) is the center of the hyperbola For any hyperbola, “2a” represents the distance along the transverse axis “2b” represents the distance along the conjugate axis “c” represents the distance from the center to either focus (the foci of a hyperbola are always along the transverse axis)

The length of the Latus Rectum for a Hyperbola is

The equations of the asymptotes for the hyperbola are these if there is a Horizontal Transverse Axis or these if there is a Vertical

Hyperbola with Transverse Axis Parallel to the x-Axis Latus Rectum

Hyperbola with Transverse Axis Parallel to the y-Axis Latus Rectum

Find an equation of a hyperbola with center at the origin, one focus at (0, 5) and one vertex at (0, -3). Determine the oblique asymptotes. Graph the equation by hand and using a graphing utility. Center: (0, 0) Focus: (0, 5) = (0, c) Vertex: (0, -3) = (0, -a) Transverse axis is the y-axis, thus equation is of the form

= 25 - 9 = 16 Asymptotes:

V (0, 3) F(0, 5) (-4, 0) (4, 0) F(0, -5) V (0, -3)

Find the center, transverse axis, vertices, foci, and asymptotes of

Center: (h, k) = (-2, 4) Transverse axis parallel to x-axis. Vertices: (h + a, k) = (-2 + 2, 4) or (-4, 4) and (0, 4)

(h, k) = (-2, 4) Asymptotes:

y - 4 = 2(x + 2) y - 4 = -2(x + 2) (-2, 8) V (-4, 4) V (0, 4) F (-6.47, 4) F (2.47, 4) C(-2,4) (-2, 0)

Hyperbolas: Example Problems Write the equation of the hyperbola with center ( 4 , -2 ) a focus ( 7 , -2 ) and a vertex ( 6, -2 )

Hyperbolas: Example Problems Find the center, foci, and graph the hyperbola:

Hyperbolas: Example Problems Find the center, foci, the length of The latus rectum, and graph the hyperbola:

Hyperbolas: Example Problems Find the center, foci, and vertices: 16x2 – 4y2 – 96x + 8y + 76 = 0

Equilateral Hyperbolas Equilateral Hyperbola: A hyperbola where a = b. When we have an equilateral hyperbola whose asymptotes are the coordinate axes, the equation of the hyperbola looks like this: xy = k. This type of hyperbola is called a rectangular hyperbola, and is easier to graph because the asymptotes are the x and y axes.  

Rectangular Hyperbolas The equation of an rectangular hyperbola is xy = k (where k is a constant value). If k >0, then your graph looks like this: If k<0, then your graph looks like this:

Rectangular Hyperbolas Example: Graph by hand the hyperbola: xy = 6.

The General form of the equation of any conic section is… Where A, B, and C are not all zero (however, for all of the examples we have studied so far, B = 0). If A = C, then the conic is a… If either A or C is zero, then we have a… If A and C have the same sign, but A does not equal C, then the conic is a… If A and C have opposite signs, then we have a ….

Conic Sections: Eccentricity Let D denote a fixed line called the directrix; let F denote a fixed point called the focus, which is not on D; and let e be a fixed positive number called the eccentricity. A conic is the set of points P in the plane such that the ratio of the distance from F to P to the distance from D to P equals e. Thus, a conic is the collection of points P for which

Conic Sections: Eccentricity 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 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 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 Find the eccentricity for the following conic section: 4y2 – 8y + 9x2 – 54x + 49= 0

Conic Sections: Eccentricity Find the eccentricity for the following conic section: 6y2 – 24y + 6x2 – 12= 0

Conic Sections: Eccentricity Write the equation of the hyperbola with center ( -3 , 1 ) focus ( 2 , 1 ) and e = 5/4

Conic Sections: Eccentricity Write the equation of an ellipse with center ( 0 , 3 ), major axis = 12, and eccentricity =2/3

Conic Sections: Eccentricity Write the equation of the ellipse and find the eccentricity, given it has foci ( 1 , -1 ) and ( 1 , 5 ) and goes through the point ( 4, 2 )

Conic Sections: Eccentricity Find the center, the foci, and eccentricity.   EX 1: 4x2 + 9y2 = 36 EX 2: 4y2 – 8y - 9x2 – 54x + 49 = 0    EX 3: 25x2 + y2 – 100x + 6y + 84 = 0

Conic Sections: Solving Systems of Equations Graphically Solve the following System of Equations by Graphing. 9x2 + 9y2 = 36 Y – 4x = 5

Conic Sections: Solving Systems of Equations Graphically Solve the following system of equations by Graphing. x2 = -4y 5x2 + y2 = 25

Conic Sections: Solving Systems of Equations Graphically Graph the following System, then state a sample solution.

Conic Sections: Solving Systems of Equations Graphically Graph the following System, then state a sample solution.

Theorem Identifying Conics without Completing the Square Excluding degenerate cases, the equation (a) Defines a parabola if AC = 0. (b) Defines an ellipse (or a circle) if AC > 0. (c) Defines a hyperbola if AC < 0.

Identify the equation without completing the square. The equation is a hyperbola.

The standard form of an equation of a circle of radius r with center at the origin (0, 0) is