The Distance Formula. The Midpoint Formula

Transcription

1 Math 120 Intermediate Algebra Sec 9.1: Distance Midpoint Formulas The Distance Formula The distance between two points P 1 = (x 1, y 1 ) P 2 = (x 1, y 1 ), denoted by d(p 1, P 2 ), is d(p 1, P 2 ) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Ex 1 Find the distance d(p 1, P 2 ) between points P 1 P 2. The Midpoint Formula The midpoint M = (x, y) of the line segment from P 1 = (x 1, y 1 ) to P 2 = (x 2, y 2 ) is M = ( x 1 + x 2 2, y 1 + y 2 ) 2 Ex 2 #28 Find the midpoint of the line segment formed by joining points P 1 P 2. P 1 = ( 10, 3); P 2 = (14,7) Ex 3 #39 Consider the three points A = ( 2, 4), B = (3, 1), C = (15, 11). a) Plot each point on a Cartesian plane form the triangle ABC. b) Find the length of each side of the triangle. c) Verify that the triangle is a right triangle. d) Find the area of the triangle. Page 1 of 9

2 Sec 9.2: Circles Conic sections are curves that result from the intersection of a right circular cone a plane. The four we ll study are below. Axis Axis Axis Axis Circle Ellipse Parabola Hyperbola Defn A circle is the set of points in the Cartesian plane that a fixed distance r from a fixed point (h, k) called the center. The fixed distance r is called the radius. The Stard Form of an Equation of a Circle (x h) 2 + (y k) 2 = r 2, where r is the radius C(h, k) is the center The General Form of the Equation of a Circle x 2 + y 2 + ax + by + c = 0 (when the graph exists) Ex 4 #12 Find the center radius of the circle. Write the stard form of the equation. Ex 5 Write the stard form of the equation of the circle with the given information. r = 8; C(h, k) = ( 4,4) Ex 6 #39 Find the center (h, k) radius r of the circle. Graph the circle. x 2 + y x + 4y + 4 = 0 Page 2 of 9

3 Ex 7 Find the stard form of each circle. a) Center at ( 2,0) containing the point (4,6). b) As Time Permits #46 Center at ( 2,3) tangent to the x-axis. Circles ( Nonlinear Systems) pg. 89 ~10 min Sec 9.3: Parabolas Defn A parabola is the collection of 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, the line D is called its directrix. In other words, a parabola is the set of points P for which d(f, P) = d(p, D). Using the distance formula, we derive: Equations of a Parabola: Vertex at (0, 0); Focus on an Axis; a > 0 Vertex Focus Directrix Equation Description (0, 0) (a, 0) x = a y 2 = 4ax (0, 0) ( a, 0) x = a y 2 = 4ax (0, 0) (0, a) y = a x 2 = 4ay (0, 0) (0, a) y = a x 2 = 4ay Parabola, axis of symmetry is the x-axis; opens to the right Parabola, axis of symmetry is the x-axis; opens to the left Parabola, axis of symmetry is the y-axis; opens up Parabola, axis of symmetry is the y-axis; opens down Parabolas with Vertex at (h, k); Axis of Symmetry Parallel to a Coordinate Axis, a > 0 Vertex Focus Directrix Equation Description (h, k) (h + a, k) x = h a (y k) 2 = 4a(x h) (h, k) (h a, k) x = h + a (y k) 2 = 4a(x h) (h, k) (h, k + a) y = k a (x h) 2 = 4a(y k) (h, k) (h, k a) y = k + a (x h) 2 = 4a(y k) Parabola, axis of symmetry parallel to x-axis; opens to the right Parabola, axis of symmetry parallel to x-axis; opens to the left Parabola, axis of symmetry parallel to y-axis; opens up Parabola, axis of symmetry parallel to y-axis; opens down Page 3 of 9

4 Ex 8 Find the vertex, focus, directrix of the parabola. Graph the parabola. x 2 = 16y Ex 9 Find the equation of the parabola described. Graph the parabola. a) #28 b) #32 Vertex at (0,0); Focus at (0,5) V(0,0); contains (2,2); axis of symmetry the x-axis Ex 10 Write an equation for the parabola. Ex 11 #40 Find the vertex, focus, directrix of the parabola. Graph the parabola. (x + 4) 2 = 4(y 1) Page 4 of 9

5 Sec 9.4: Ellipses Defn An ellipse is the collection of all points in the plane such that the sum of the distances from two fixed points, called the foci, is a constant. Use cardboard, string, thumbtacks to physically construct an ellipse. The length of string is the constant, 2a. The line containing the foci is called the major axis. The midpoint of the line segment joining the foci is called the center. The line through the center perpendicular to the major axis is called the minor axis. The two points of intersection of the ellipse the major axis are the vertices, V 1 V 2, of the ellipse. The distance from one vertex to the other is called the length of the major axis. Note: We still study only ellipses whose major axis is parallel to (or coincides with) either the x- or y axis. Foci V 1 Major axis y Center P = (x, y) V 2 Length of major axis: x Minor axis Ellipses with Center at the Origin Center Major Axis Foci Vertices Equation (0,0) x-axis (0,0) y-axis ( c, 0) (c, 0) (0, c) (0, c) ( a, 0) (a, 0) (0, a) (0, a) x 2 a 2 + y2 b 2 = 1 a > b b 2 = a 2 c 2 x 2 b 2 + y2 a 2 = 1 a > b b 2 = a 2 c 2 Ellipses with Center (h, k) Major Axis Parallel to a Coordinate Axis Center Major Axis Foci Vertices Equation (x h) (h + c, k) (h + a, k) 2 (y k)2 (h, k) Parallel to x-axis (h c, k) (h a, k) a 2 + b 2 = 1 a > b b 2 = a 2 c 2 (h, k) Parallel to y-axis (h, k + c) (h, k c) (h, k + a) (h, k a) (x h) 2 (y k)2 b 2 + a 2 = 1 a > b b 2 = a 2 c 2 Ex 12 #24 Find the vertices foci of the ellipse. Graph the ellipse. 9x 2 + y 2 = 81 Page 5 of 9

6 Ex 13 Find an equation for each ellipse graph each one. a) #26 b) #32 C(0,0); F(2,0); V(5,0) Foci at (±6,0); length of major axis is 20 Ex 14 #42 Graph the ellipse. 16x 2 + 9y 2 128x + 54y 239 = 0 Sec 9.5: Hyperbolas Defn A hyperbola is the collection of all points in the plane the difference of whose distances from two fixed points, called the foci, is a positive constant. The line containing the foci is the transverse axis. The midpoint of the line segment joining the foci is the center of the hyperbola. The line through the center perpendicular to the transverse axis is the conjugate axis. The hyperbola consists of two separate curves called branches. The two points of intersection of the hyperbola transverse axis are the vertices, V 1 V 2, of the hyperbola. The positive constant = Page 6 of 9

7 Hyperbolas with Center at the Origin Center Transverse Axis Branches Open Foci Vertices Equation Asymptotes (0,0) x-axis Left right (0,0) y-axis Up down ( c, 0) (c, 0) (0, c) (0, c) ( a, 0) (a, 0) (0, a) (0, a) Ex 15 Find the equation of the hyperbola described. Graph the equation. Plot 3 points per branch or use asymptotes. a) #26 b) #27 Vertices at (0,6) (0, 6); Focus at (0,8) F( 10,0); F(10,0); V( 7,0) x 2 a 2 y2 = 1, where b2 b 2 = c 2 a 2 or c 2 = a 2 + b 2 y 2 a 2 x2 = 1, where b2 b 2 = c 2 a 2 or c 2 = a 2 + b 2 y = ± b a x y = ± a b x Ex 16 #16 x 2 16 y2 64 = 1 Graph the equation using its asymptotes. Page 7 of 9

8 Ex 17 Find the equation of the hyperbola described. Graph the equation. a) #36 b) #37 V(0, ±4); asymptote the line y = 2x F(±3,0); asymptote the line y = x Sec 9.6: Systems of Nonlinear Equations In a system of nonlinear equations, at least one equation is nonlinear. Ex 18 Solve the system of nonlinear equations using substitution. a) #8 b) #9 { y = 100 x2 { x2 + y 2 = 4 x + y = 14 y = x 2 2 Page 8 of 9

9 Ex 19 #19 Solve the system of nonlinear equations using elimination. { x2 2x y = 8 6x + 2y = 4 Ex 20 Solve the system of nonlinear equations using the method you wish. { 9x2 + 4y 2 = 36 x 2 + (y 7) 2 = 4 (Circles ) Nonlinear Systems pg. 90 ~10 min Page 9 of 9