12.5 Ellipses and Hyperbolas

Conic Sections Hyperbola Parabola Ellipse Circle

General Form Equation of Conic Sections Where ‘A’, ‘B’, ‘C’, ‘D’, ‘E’, and ‘F’ are real numbers AND ‘A’, ‘B’, and ‘C’ are not all zero.

Vocabulary Ellipse The set of all points in a plane whose distance from two fixed points in the plane have a constant sum. P(x,y) d1 d2 x-axis Center (0, 0)

Vocabulary Ellipse The set of all points in a plane whose distance from two fixed points in the plane have a constant sum. P(x,y) d1 c d2 x-axis Center (0, 0) Distance from (-c, 0) to (x, y) + distance from (c, 0) to (x, y) = 2a

Vocabulary Ellipse The set of all points in a plane whose distance from two fixed points in the plane have a constant sum. P(x,y) d1 c d2 x-axis Center (0, 0)

Ellipses Center at (0, 0) Major axis: longest axis Standard Form Equation of an ellipse. Minor axis: shortest axis y-axis a a x-axis Vertex (a, 0) Vertex (-a, 0) Center (0, 0)

Ellipses Center at (0, 0) Major axis: longest axis Standard Form Equation of an ellipse. Minor axis: shortest axis y-axis Center (0, 0) b x-axis b

Vocabulary Foci: (plural of focus) the fixed points from which the sum of the distances from these two points to any point on the ellipes is a constant value. Focal Axis: the axis that passes thru the 2 foci. Major Axis: is the same as the focal axis. It is the longest axis. It is either the x-axis or the y-axis. Minor Axis: an axis that splits the ellipse into symetrical halfs and is perpedicular to the major axis. Vertexes: The points on the ellipse on the main axis.

Ellipse with Vertex (0, 0) Focal Axis x-axis y-axis: Standard Equations: Foci: Vertexes: Semi Major Axis: a a Semi Minor Axis: Pythagorean relation:

Example Problem: Standard Equations: Opening Direction: down Focus: (0, -1/8) Directrix: Axis: y-axis y-axis Focal Length: Focal width:

Your turn: for the following equation, find the following information about the parabola. 1. Opening Direction: 2. Focus: 3. Directrix: 4. Axis: 5. Focal Length: 6. Focal width:

Example Problem: Standard Equations: Opening Direction: Left/right Focus: (p, 0) Directrix: Axis: x -axis Focal Length: Focal width:

Your turn: 7. Find the equation of the parabola that has a directrix of y = -4 and a focus of (0, 4).

Parabolas with Vertex (h, k) Standard Equations: Opening Direction: Up/down Right/Left Focus: (h, k + p) (h + p, k) Directrix: Axis: x = h y = k Focal Length: Focal width:

Example Problem: y value of vertex and y value of focus are the same  axis is a horizontal line. Standard Equations: (5, 4) Opening Direction: Left/right right Focus: (h + p, k) (3 + p, 4) = (5, 4) (3, 4) p = 2 Directrix: x = 3 - 2 x = 1 Axis: y = k y = 4 Focal Length: 2 8 Focal width:

Your turn: 7. Find the equation of the parabola that has a vertex of (2, 5) and a focus of (2, -2). Vertex: (h, k) h = 2 k = 5 Focus below vertex: opens down Focus: (h, k + p) = (2, -2)  5 + p = -2 p = -7

General Form Equation of Conic Sections Parabolas: Ellipses:

Ellipses Center at (0, 0) Ellipse: the set of all points whose distance the center is a constant sum. (0, b) x-axis Vertex (a, 0) Vertex (-a, 0) Focus (-c, 0) Center (0, 0) Focus (c, 0) (0, -b)

Ellipses Center at (0, 0) Switch the positions of the vertex and focus. (0, b) x-axis Vertex (a, 0) Vertex (-a, 0) Focus (-c, 0) Center (0, 0) Focus (c, 0) (0, -b)

Hyperbola Center at (0, 0) Switch the positions of the vertex and focus. (0, b) x-axis Vertex (-a, 0) Center (0, 0) Focus (c, 0) Focus (-c, 0) Vertex (a, 0) (0, -b)

Hyperbola Center at (0, 0) The minor axis endpoints (0, b) and (0, -b) no longer apply. (0, b) x-axis Vertex (-a, 0) Center (0, 0) Focus (c, 0) Focus (-c, 0) Vertex (a, 0) (0, -b)

Hyperbola Center at (0, 0) The set of all points whose distance from the center is a constant difference. Point (x, y) x-axis Center (0, 0) Focus (c, 0) Focus (-c, 0) Vertex (-a, 0) Vertex (a, 0)

Deriving the Equation for a Hyperbola. substitution Divide both sides by Divide both sides by You don’t have to derive this. You should know that the points that come from the equation are all the points whose distance from two fixed points in the plane have a constant difference.

Ellipse Hyperbola. Wow: one is addition, the other is subtraction. Ellipse: the set of all points whose distance (from two fixed points) is a constant sum. Hyperbola: the set of all points whose distance (from two fixed points) is a constant difference.

Pythagorean Relation for: Ellipse Hyperbola.

Vocabulary Foci: (plural of focus) the fixed points from which the sum of the distances from these two points to any point on the hyperbola is a constant value (equal to the transverse axis). Transverse Axis: the segment connecting the two vertexes  length = 2a Semi –Transverse Axis:  length = a Conjugate Axis: the line segment whose midpoint is the center point of the hyperbola and which is perpendicular to the transverse axis.  length = 2b Semi –conjugate Axis:  length = b

Your turn: 1. What is the equation of the hyperbola centered at the origin whose transverse axis is on the x-axis and whose semi-transverse axis is 4 units long and whose semi-conjugate axis is 3 units long?

Hyperbola Center at (0, 0) x-axis Has two asymptotes Vertex (-a, 0) Focus (-c, 0) Focus (c, 0)

Hyperbola Center at (0, 0) x-axis Has two asymptotes Vertex (-a, 0) Conjugate Axis Center (0, 0) Vertex (-a, 0) x-axis Vertex (a, 0) Focus (-c, 0) Focus (c, 0) Transverse Axis

Hyperbola with Center (0, 0) Focal Axis x-axis y-axis: Standard Equations: Foci: Length Semi Transverse Axis: a a Vertexes: Length Semi Conjugate Axis: Pythagorean relation: (used to find ‘c’) Asymptotes:

Find the equation of the hyperbola. Center: (0, 0) Semi-transverse axis: (y-axis) length = 5 Semi-conjugate axis: 3  b = 3  a = 5 Standard equation is of the form: Find the foci of a hyperbola.

Your turn: for the following equation, find the following information about the hyperbola. 2. Vertexes: 3. Length of the semi-transverse axis. 4. Length of the semi-conjugate axis. 5. Foci: 6. The asymptotes

Your turn: 7. Find the equation of the hyperbola from the following information: Foci: (3, 0), (-3, 0) Length transverse axis: 4

Hyperbola with Center (h, k) Standard Equations: Focal Axis y = k x = h Foci: Length Semi transverse Axis: a a Vertexes: Length Semi conjugate Axis: Pythagorean relation: (used to find ‘c’) Asymptotes:

Example Find the equation of the hyperbola with the following information given: Vertexs: (1, -4), (1, 8)  Semi-transverse axis is the line x = 1  Semi-transverse axis is vertical: equation of hyperbola is: Center  (1, 2) h =1, k = 2  Semi-transverse axis length = 8 – (-4) = 12  a = 6 Semi-conjugate axis length: 4  b = 4

Your turn: for the following equation, find the following information about the hyperbola. (Hint: put it into standard form) 8. Transverse axis: (horizontal or vertical?) and what is its equation? 9. What is the center of the hyperbola. 10. Length of the semi-conjugate axis. 11. Ordered pairs for the vertexes. 12. The ordered pairs for the Foci (use the Pythagorean relationship) 13. Length of the semi-transverse axis.

Your turn: 14. Find the equation of the hyperbola with the following information given: Vertexes (1, -4) and (5, -4) Foci: (0, -4) and (6, -4)

Vocabulary: Eccentricity: the shape of the hyperbola is related to its eccentricity. The more eccentric the hyperbola is, the more “squished” it appears to be.