Conic Sections There are 4 types of Conics which we will investigate: 1.Circles 2.Parabolas 3.Ellipses 4.Hyperbolas

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

Graph ()()xy  by hand.

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

Find the center, the radius and graph: x 2 + y x – 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

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. y 2 – 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. x 2 + 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. x 2 – 4x – 12y – 32 = 0 Parabolas: Example Problems

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

Find the vertex, focus anddirectrix of xxy 

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

(-2, -3) (-2, -1) y = -5 (-6, -1) (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. y x P = (x, y) Focus 2 Focus 1 Major Axis Minor Axis

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

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

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: 16x 2 + 4y 2 – 96x + 8y + 84 = 0 Ellipses: Example Problems

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

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

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) (0, -7) FOCI

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) =

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

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

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

The length of the Latus Rectum for a Hyperbola is

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

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: Hyperbolas: Example Problems 16x 2 – 4y 2 – 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 ….

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: 4y 2 – 8y + 9x 2 – 54x + 49= 0

Conic Sections: Eccentricity Find the eccentricity for the following conic section: 6y 2 – 24y + 6x 2 – 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: 4x 2 + 9y 2 = 36 EX 2: 4y 2 – 8y - 9x 2 – 54x + 49 = 0 EX 3: 25x 2 + y 2 – 100x + 6y + 84 = 0

Conic Sections: Solving Systems of Equations Graphically Solve the following System of Equations by Graphing. 9x 2 + 9y 2 = 36 Y – 4x = 5

Conic Sections: Solving Systems of Equations Graphically Solve the following system of equations by Graphing. x 2 = -4y 5x 2 + y 2 = 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.

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).

C(-2,4) (-4, 4) (0, 4) (-2, 8) (-2, 0) Write the equation for this hyperbola:

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.

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

If e = 1, the conic is a parabola. If e < 1, the conic is an ellipse. If e > 1, the conic is a hyperbola.

r Directrix D Pole O (Focus F) p d(D,P)

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.

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

TheoremEquation of an Ellipse; Center at (0, 0); Foci at (+ c, 0); Major Axis along the x-Axis An equation of the ellipse with center at (0, 0) and foci at (- c, 0) and (c, 0) is The major axis is the x-axis; the vertices are at (-a, 0) and (a, 0).

y x F 2 =(c, 0) F 1 =(-c, 0) V 2 =(a, 0) V1V1 (0, b) (0, -b) = (-a, 0)

TheoremEquation of an Ellipse; Center at (0, 0); Foci at (0, + c); Major Axis along the y-Axis An equation of the ellipse with center at (0, 0) and foci at (0, - c) and (0, c) is The major axis is the y-axis; the vertices are at (0, -a) and (0, a).

y x V 2 = (0, a) V 1 = (0, -a) (b, 0) ( -b, 0) F 2 = (0, c) F 1 = (0, -c)

TheoremEquation of a Hyberbola; Center at (0, 0); Foci at ( + c, 0); Vertices at ( + a, 0); Transverse Axis along the x-Axis An equation of the hyperbola with center at (0, 0), foci at ( - c, 0) and (c, 0), and vertices at ( - a, 0) and (a, 0) is The transverse axis is the x-axis.

TheoremEquation of a Hyberbola; Center at (0, 0); Foci at ( 0, + c); Vertices at (0, + a); Transverse Axis along the y-Axis An equation of the hyperbola with center at (0, 0), foci at (0, - c) and (0, c), and vertices at (0, - a) and (0, a) is The transverse axis is the y-axis.

Theorem Asymptotes of a Hyperbola The hyperbola has the two oblique asymptotes

Theorem Asymptotes of a Hyperbola The hyperbola has the two oblique asymptotes

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

= = 16 Asymptotes:

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

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)

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

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

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

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

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

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

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

Graph ()()xy  using a graphing utility.

The LATUS RECTUM is a line segment, either horizontal or vertical, that joins the focus with two points on the parabola. The length of the Latus Rectum = 4a. Parabolas: Example Problems

Latus Rectum: Let y = -1

Ellipse with Major Axis Parallel to the x-Axis where a > b and b 2 = a 2 - c 2. (h - a, k)(h + a, k)(h, k) (h - c, k) (h + c, k) y x Major axis

Ellipse with Major Axis Parallel to the y-Axis where a > b and b 2 = a 2 - c 2. y x (h, k + a) (h, k - a) (h, k) (h, k + c) (h, k - c) Major axis

Hyperbola with Transverse Axis Parallel to the y-Axis; Center at (h, k) where b 2 = c 2 - a 2.

Hyperbola with Transverse Axis Parallel to the x-Axis; Center at (h, k) where b 2 = c 2 - a 2.

Conic Sections Part I: CIRCLES

D: x = -a F = (a, 0) x y V

D: x = a F: (-a, 0) V y x

y x D: y = -a V F: (0, a)

x y D: y = a F: (0, -a)

Lesson Overview 10-8A

Lesson Overview 10-8B