Distance and Midpoint Formula 7.1

Transcription

1 Distance and Midpoint Formula 7.1 Distance Formula d ( x - x ) ( y - y ) 1 1 Example 1 Find the distance between the points (4, 4) and (-6, -). Example Find the value of a to make the distance = 10 units given the points (-7, 3) and (a, 11). Example 3 Show that M(, 4) is the midpoint of the segment joining A(7, ) and B(-3, 6). Midpoint (Line Segment) x + x y, + y 1 1 Example 4 Find the center of a segment whose coordinates are A(-, 3) and B(8, -5). Example 5 Circle x has a diameter MN. If M is at (-4, ) and the center is (-6, 3), find the coordinates of N. Pg 411, -34 even

2 Parabolas 7. Conic Sections any figure that can be formed by slicing a double cone. Parabola the set of all points in a plane that are the same distance from a given point called the focus and a given line called the directrix. Information about Parabolas Form of Equation y a( x - h) k x a( y - k) h Axis of Symmetry x = h y = k Vertex (h, k) (h, k) Focus 1 1 h, k h, k 4a 4a Directrix 1 1 y = k - x h - 4 a 4 a Direction of Opening a(+): Up, a(-): Down a(+): Right, a(-): Left Length of Latus 1 a 1 a Example 1 Graph y x 1 x 14

3 Example Graph 5 x y 4 y - 6 Pg 419, 6-1, 16-1,5,8

4 Parabolas 7. (Day ) Writing Equations Need a and vertex to write equation Remember: 1. Focus, Directrix, and Latus all have a as part of their formula.. a and Latus are reciprocals of one another. Example 3 Write the equation of the cross-section of a satellite dish with focus units from the vertex and a latus 8 units long. Assume that the focus is at the origin and the parabola opens to the right. Example 4 Write and equation for the parabola shown below. Example 5 Focus = (-4, -) and Directrix: x = -8 Pg 419, 30-43

5 Circles 7.3 Circle the set of all points in a plane that are equidistant from a given point in the plane, called the center. Equation of a Circle ( x - h) ( y - k) = r Center: (h, k) Radius: r Example 1 Write an equation of a circle that has a radius 15 and a center (-9, -6) and then graph. Example Find the center and radius of circle with equation x x y + 4 y and then graph. Tangent a line that intersects a circle at exactly one point. Example 3 Write an equation of the circle that has its center at (, -4) and is tangent to the x-axis. Example 4 Write an equation of a circle if the endpoints of a diameter are at (1, 8) and (1, -6). Pg 46, 1-49 odds

6 Ellipses 7.4 Ellipse the set of all points in a plane such that the sum off the distances from the foci is constant. **An ellipse has axis of symmetry** c + b = a Major axis = a b = a - c Minor axis = b Standard Equations of Ellipses with center at origin x y x y + = 1 OR + = 1 a b b a ***the largest value is always a*** Example 1 Find the coordinates of the foci and the lengths of the major and minor axes of an ellipse whose equation is 49 x + 16 y = 784 and then graph.

7 Example Find the coordinates of the foci and the lengths of the major and minor axes of an ellipse whose equation is 5 x + 36 y = 900 and then graph. Writing Equations Need a and b Example 3 Write the equation of the ellipse shown below. Pg 436, 6, 8, 10, 14-16, 19, 0, 3-6, 35

8 Example 4 Graph ( x + 1) ( y + ) + = Ellipses 7.4 (Day ) Standard Equations of Ellipses with center (h, k) ( x - h) ( y - k) ( x - h) ( y - k) + = 1 OR + = 1 a b b a Example An equation of an ellipse is 16 y + 9 x - 96 y - 90 x = - 5. Find the coordinates of the center, foci, the lengths of the major and minor axes, and then graph. Pg 436, 7, 9, 11-13, 17, 18, 1,, 7-3, 34, 36

9 Hyperbolas 7.5 Hyperbola the set of all points in a plane such that the absolute value of the difference of the distances from any point on the hyperbola to two given points, called the foci, is constant. (a) a (b) + b = c Standard Equations of Hyperbolas with Center at Origin x y y x - = 1 OR - = 1 a b a b **when x is positive, the transverse axis is horizontal; when y is positive, the transverse axis is vertical.** Equation of Hyperbola Equation of Asymptote Transverse Axis x y - a b = 1 y = b x a Horizontal y x - a b = 1 y = a x b Vertical *Positive value is where the transverse axis lies*

10 Example 1 x y A comet travels along a path that is one branch of a hyperbola whose equation is - = coordinates of the vertices, foci, equations of the asymptotes, and then draw the figure.. Find the Example Write an equation for the hyperbola 6 y - 3 x = 4 in standard form and graph. Example 3 Write an equation of a hyperbola with a foci at (0, 7) and (0, -7) if the length of the transverse axis is 6 units. Remember, when writing equations we need a and b. Pg 445, 5-8, 10, 11, 14-16, 19-, 37

11 Hyperbolas 7.5 (Day ) Standard Equations of Hyperbolas with Center at (h, k) ( x - h) ( y - k) ( y - k) ( x - h) - = 1 OR - = 1 a b a b **when x is positive, the transverse axis is horizontal; when y is positive, the transverse axis is vertical.** Example 1 Draw the graph of ( y + ) ( x - 3) - = Example Write the equation of the Hyperbola shown below. Example 3 The graph of 144 y - 5 x y x = 349 is a hyperbola. Find the standard form of the equation, coordinates of the vertices and foci, the equations of the asymptotes and draw. Pg 445, 9, 1, 13, 18, 3-5, 35, 36

12 Conic Sections 7.6 Equation of a Conic Section: Ax + Bxy + Cy + Dx + Ey + F = 0 Conic Section Standard Form of Equation Relationship of A and C Parabola y = a( x - h) + k or x = a( x - k) + k A = 0 or C = 0, not both Circle y = ( x - h) + ( y - k) = r A = C Ellipse ( x - h) ( y - k) ( x - h) ( y - k) A and C have the same + = 1 OR + = 1 a b b a sign and A C Hyperbola ( x - h) ( y - k) ( y - k) ( x - h) A and C have - = 1 OR - = 1 opposite signs a b a b Example 1 Identify 9 x + 16 y - 54 x + 64 y + 1 = 0 and graph. Example Identify y - 3 x + 6 y + 1 = 0 and graph. Example 3 Identify x - 14 x + 4 = 9 y - 36y and graph. Pg 453, 5-31, 41, 43 odd

13 Solving Quadratic Systems 7.7 Example 1 Solve. y = ( x - ) + 1 y = -4 x + 5 Algebra Graphing Conic Sections Example Solve 4 x - y = 36 ( x - 5) + y = 64 Algebra Graphing

14 Example 3 Solve algebraically. x + y = 9 5 x - y = 0 Example 4 Solve by Graphing. 10 ( x - 5) + y y - x + 9 Pg 464, 7-37 odd