Flash Light Reflectors. Fountains and Projectiles. Algebraically, parabolas are usually defined in two different forms: Standard Form and Vertex Form

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2 Sec 6.1 Conic Sections Parabolas Name: What is a parabola? It is geometrically defined by a set of points or locus of points that are equidistant from a point (the focus) and a line (the directrix). To see this idea visually try drawing a straight line at the bottom of a piece of paper with a ruler. Then, place a point in the middle just above the line as shown. Next, fold the paper multiple times in various locations so that the line folds on top of the point and make a crease. The creases will outline a parabola. Why does this happen? Parabolas can be found in many places in everyday life. A few examples are shown below. Can you guess where the focus point should be in the flash light or the satellite dish? Fountains and Projectiles Flash Light Reflectors Algebraically, parabolas are usually defined in two different forms: Standard Form and Vertex Form Let s start with the most basic graph of a parabola, Satellite Dishes M. Winking (Section 6-1) p.99

3 So, where would the focus point and the directrix be in the basic equation of y = x. This is not a trivial task. We know that the directrix should be somewhere below the vertex and the distance from the vertex to the line should be the same as the distance from the vertex to the focus point. A (x,x ) Consider putting an arbitrary point on the y axis above the vertex and call it the focus point at (0, p). We will need to figure out what p is to find the location of the focus. Then, we know that the directrix is the same distance away from the vertex but on the other side of the parabola and it is a line. So, it the directrix would need to be the line at y = p. y = p The parabola is geometrically defined as the set of point equidistant from a point and a line. So, we already know algebraically the parabola is given by the equation y = x which would suggest every point on the parabola is basically of the form (x, x ) (eg. (1,1), (, 4), (3,9), etc.). We described the point of the focus point as (0,p) and then, if you think about it carefully the point directly below the general point of the parabola shown in the diagram on the directrix would have to be the point (x, p). Now, we can just use the distance formula to say that AF = AB. ( x x ) + ( y y ) = ( x x ) + ( y y ) A F A F A B A B (0,p) F B (x,-p) ( ) ( ) ( ) x 0 + x p = x x + ( x p) We can square both sides: ( ) ( ) ( ) x 0 + x p = x x + ( x p) Clean things up a bit: ( ) 0 ( ) x + x p = + x + p Expand (use F.O.I.L. if necessary): 4 4 x + x x p + p = x + x p + p Cross out items on both sides (subtract from both sides) x + x 4 So the focus is located at x p + p = 4 x + x p + p Move the x p to the right: x x p = x p x + x p + x p = 4x p Solve for p x 4x p = 4x 4x Which reduces 1 = p 4 1 0, 4 and the directrix is located at 1 y = 4 M. Winking (Section 6-1) p.100

4 It turns out doing nearly the same thing for the vertex form of the parabola, ( y k ) = a ( x h) OR ( x h) = a ( y k ) we can show in a similar manner that the focus is located inside the mouth of the parabola a vertical distance of 1 from the vertex whereas the directrix is located the same distance away from the vertex on the other 4a side of the parabola from the focus. 1. Sketch a graph the following Parabolas (Label the Vertex, Focus, and Directrix) a. ( y ) = 1( x+ 3) b. ( y+ 1) = 1 ( x 4) 8 x+ 3 = 1 y 4 c. ( y 4) = ( x ) d. ( ) ( ) M. Winking (Section 6-1) p.101

5 x+ 4 = 1 y f. 1 e. ( ) ( ) y = 8x. Find each of the requested components of a parabola. a. Given the vertex of a parabola is located at (,) and has a directrix of y = 4, find the location of the focus. b. Given the vertex of a parabola is located at (,1) and has a focus at ( 1,1), find the equation of the directrix. c. Given the directrix of a parabola is y = 3 and a focus located at (4,1), find the location of the vertex. Which way does the parabola open? d. Find the vertex of the parabola defined by y x x = 6 + M. Winking (Section 6-1) p.10

6 3. Find each of the requested components of a parabola. a. Given the vertex of a parabola is located at (-, -3) and has a directrix of y = -6, find the location of the focus and the equation of the parabola in standard form. b. Given the focus of a parabola is located at (1.5,1) and has a directrix at x =.5, find the coordinates of the vertex and the equation of the parabola in standard form.. c. Find the vertex, directrix, and focus of the following parabola defined by: y + 1 = x + 4x d. Find the vertex, directrix, and focus of the following parabola defined by: x = y 4y + 1 M. Winking (Section 6-1) p.103

7 1. Sec 6. Conic Sections Circles Name: What is a circle? It is geometrically defined by a set of points or locus of points that are equidistant from a point (the center). Consider the circle at the right. What is the length of every segment drawn from center O to a point on the edge of the circle? How would you find the length of segment OB? 1. Basic Circles Graph the following: A. + y = 4 x B. x + y = 36 C. x + y = 0. Translated Circles Graph the following: A. ( x 4) + ( y + ) = 9 B. ( x + 3) + ( y + 1) = 5 C. ( x + ) + ( y 3) = 18 M. Winking Unit 6- page 104

8 3. Equations of Circles Find the equation of each of the following: a. ( x ) + ( y ) = b. ( x ) + ( y ) = 4. Equations of Circles The following design is composed of 3 full circles and semi-circles. Can you find the equations of each and put them in your calculator? a. ( x ) + ( y ) = b. ( x ) + ( y ) = c. ( x ) + ( y ) = d. ( x ) + ( y ) = e. ( x ) + ( y ) = ** When you put these in your TI-83/84 calculator you will have to solve for y using the square root method you may have to use two equations to describe a complete circle. For example, if you wanted to graph the 1 complete circle ( x 4) + ( y + ) = 9. It would require that you use two equations = 9 ( x 4) ( 4) y = 9 x y and M. Winking Unit 6 - page 105

9 5. Finding Standard form of circles. Put the following circles in standard form and graph them. A. + y 10y + 9 = 0 x B. x + y x + 8y = 8 C. + y 6y x = 15 x D. x + y 8x + 1y = 6 M. Winking Unit 6- page 106

10 6. Find the Equations of the Following Circles in Standard form. A. If the endpoints of a diameter of a circle are A:( 4, ) and B:(, ), what is the equation of the circle. B. If the endpoints of a diameter of a circle are A:( 3, ) and B:(7, 4), what is the equation of the circle. C. If the endpoints of a radius of a circle are A:(,3) and B:(6,1) with the center at point A, what is the equation of the circle. B A B A B A ( x ) + ( y ) = ( x ) + ( y ) = ( x ) ( y ) 7. Applications of circles. A. A new company E-Mobile is setting up Ad based free Wi-Fi Access Points. Each mobile tower that E-Mobile creates covers the area of a circle with a radius of 4 miles with the tower at the center. Sketch the following circles that show on the map the coverage areas. ( x+ 7) + ( y 4) = 16 ( x ) ( y ) ( x+ 6) + ( y+ 8) = 16 ( ) x y = 16 + = 16 B. A Bus travels the entire length of U.S. 9 and stops at the bust stops located at A:( 8, 6), B:( 4, 3), C:(4, 1), D:(8, 3). Which bus stops have access to the free Wi-Fi Access by E-mobile? + = 4 miles Gwinnett County, GA C. A company will be building an Urgent Care facility in Pickens County. Ideally, the company would like to be within 6 miles reach of the most citizens. Create an equation of a circle that encapsulates as many cities as possible using a 6-mile radius and the center being where the Urgent Care facility should be created. 4 miles ( x ) ( y ) + = M. Winking Unit 6- page 107

11 1. Sec 6.3 Conic Sections Ellipses Name: An ELLIPSE could be accurately described as circle that has been stretched or compressed by a constant ratio towards a diameter of a circle. A circle is actually a special type of ellipse. More explicitly, an ELLIPSE is the locus of points whose distance from two focal points is constant. To better understand what this means consider the following: Paste or tape a coordinate grid to a piece of cardboard. Put two push pins in the grid at ( 4, 0) and (4, 0). These pins will represent the focal points. Next tie a string to each push pin such that the length of the string between the two pins is approximately 10 units long (using the unit length of the coordinate grid). Finally, take a pencil and stretch out the string until the string is straightened and trace out all of the places the string allows the pencil to move with the string remaining taut. This will ensure that the combined distance from each focal point will always be a total of 10 units long. Rather than always having a fixed diameter an oblong ellipse can be described as having a major axis (the longest diameter) and a minor axis (the shortest diameter). The endpoints of the major axis are considered the vertices of an ellipse and the endpoints of the minor axis are consider the co-vertices. 1. Determine the length of the major & minor axis. List the coordinates of vertices and co-vertices of the following ellipses. A. Major Axis Length: B. Major Axis Length: Minor Axis Length: Vertices: Minor Axis Length: Vertices: Co-Vertices: Co-Vertices: M. Winking Unit 6-3 page 108

12 Rather than always having a fixed radius an oblong ellipse can be described as having a major radius/semi-major axis(the longest radius) and a minor radius/semi-minor axis (the shortest radius). The major radius is commonly denoted by the variable a and the minor radius is commonly denoted by the variable b. These different radii are used to write the equation of the ellipse in standard form. Standard Form of an ELLIPSE centered at the origin. x y + a = 1 OR x b b y + a = 1 ** a must always be the largest radius ** The Equation of the ELLIPSE shown at the left. x y = 1 OR x 5 + y 9 = 1 Standard Form of an ELLIPSE (x h) a + (y k) b = 1 OR (x h) b + (y k) a = 1 ** a must always be the largest radius ** The Equation of the ELLIPSE shown at the left. (x ( 3)) + (y 1) 3 = 1 OR (x+3) 4 + (y 1) 9 = 1. Determine the equations of the following ellipses in standard form (list the vertices and co-vertices). A. B. C.

13 3. Graph the following equations and list the center and radius of each ellipse. A. x 5 + y 4 = 1 B. (x ) 9 + (y+3) 1 = 1 C. 4(x 1) + 9(y ) = Complete the square to put each of the following ellipses in standard form. Then, list the center, vertices, and co-vertices. Finally, graph each ellipse. 16x + 4y 3x = Complete the square to put each of the following ellipses in standard form. Then, list the center, vertices, and co-vertices. Finally, graph each ellipse. 9x + y + 54x 4y = 76 M. Winking Unit 6-3 page 110

14 Finding the focal points algebraically, requires the use of the Pythagorean Theorem. First, consider the constant distance found between any point on the ellipse and the two focal points is equal to the length of the major axis. In the above example, you can see that m + n = 10 which is also the length of the major axis. Now, if we position the string at one of the co-vertices then a right triangle is formed which is shown below. The hypotenuse is half the length of the major axis which would equal to the length of the major radius. One of the legs of the right triangle is the minor radius and the other leg is represented by the distance from the center of the ellipse to a focal point. It may seem a little out of order but you can see based on the defined variables we could state the following. The focal points are always located on the major axis and are a distance of c units away from the center of the ellipse where c is defined by the equation: b + c = a OR a b = c ** a is the length of the major radius and b the minor radius ** These equations correspond to the ellipse shown at the left = 5 OR 5 3 = 4 6. Find the focal points of each ellipse shown below. A. B. M. Winking Unit 6-3 page 111

15 Eccentricity of an Ellipse is a measure of how close the ellipse is to being a circle and is given by the formula: Eccentricity = c a, where c is the distance from the center to a focal point and a is the length of the major radius. Sample Ellipses Eccentricity = 0 Eccentricity 0.75 Eccentricity 0.94 Eccentricity Find the eccentricity of each of the following ellipses also used in problem number 6. A. B. 8. Find the equations of ellipse given the following parameters and sketch a graph. A. Find the equation and graph of an ellipse that has vertices at (-, 5) and (-, -3) and co-vertices at (-4,1) and (0,1). B. Find the equation and graph of an ellipse that has vertices at (-6, 1) and (4, 1) and focal points at (-5,1) and (3,1). M. Winking Unit 6-3 page 11

16 9. Consider the following describes the orbit of Halley s Comet. It orbits the sun in a highly elliptical orbit with the sun being one of the focal points of the ellipse. Determine the following given that a = astronomical units and b = astronomical units. Halley s Comet A. Determine an equation that describes the orbit of Halley s comet. B. Find the position of the Sun on the coordinate grid. C. What is the Eccentricity of the orbit? 10. In a Introduction to Theater Course, a student noticed that the spotlight actually cast a spotlight on the stage floor in the shape of an ellipse. The student laid out a coordinate grid on the floor. The units are in feet. A. What is the equation of the ellipse created by the spotlight on the coordinate grid? B. What are the coordinates of the focal points? C. What is the eccentricity of the ellipse? M. Winking Unit 6-3 page 113

17 1. Sec 6.4 Conic Sections Hyperbolas Name: A HYPERBOLA is the locus of points whose difference in distance from two focal points is constant. To better understand what this means consider the following example of the graph of x y = 1 with focal points at ( 5, 0) and (5, 0) Example 1 Example P:(5,.5) P:(6, 3.4) d 1 d = difference in distance from the focal points = 8 d 1 d = difference in distance from the focal points = 8 Notice, that the distance remains a constant 8 units which is also equal to the distance between the two vertices or a. As shown the graph at the right. The distance from the left focal point to the vertex on the right is 9 units. The distance from the right focal point to the vertex on the right is 1 unit. So, the difference is again 8 units. P:(4, 0) = 8 Next let s look at the graph shown to the left of with some descriptions of points and axes. x 16 y 9 = 1 M. Winking Unit 6-4 page 114

18 Determine the equations of the following Hyperbolas in standard form (list the vertices, co-vertices, and the equations of the asymptotes) Graph the following equations and list the center and radius of each hyperbola (label the vertices and co-vertices) 4. y 1 x 9 = 1 5. (x+) 4 (y 1) 4 = 1 6. (y ) 1 x 9 = 1 M. Winking Unit 6-4 page 115

19 7. Complete the square to put the following hyperbola in standard form. Then, list the center, vertices, and co-vertices. Finally, graph the hyperbola. 4x 9y 16x = 0 8. Complete the square to put the following hyperbola in standard form. Then, list the center, vertices, and co-vertices. Finally, graph the hyperbola. y 9x + 18x 4y = 14 M. Winking Unit 6-4 page 116

20 Given a HYPERBOLA of the form x y a b = 1 the focal points are located a distance of c units away from the center on the transverse or major axis (i.e. in this example at ( c, 0) and (c, 0) given the relationship a + b = c. The proof of this claim is an optional exploration for this course. Consider the following diagram. Use the distance formula to describe d 1 and d. d1 = d = Next, substitute those descriptions in to the statement below and simplify a lot. We discussed earlier that the constant difference in distance is always equal to a. d1 d = a M. Winking Unit 6-4 page 117

21 Given a HYPERBOLA of the form x y a b = 1 the focal points are located a distance of c units away from the center on the transverse or major axis (i.e. in this example at ( c, 0) and (c, 0) given the relationship a + b = c. Using this relationship find the focus points for the following hyperbolas x 9 y 1 = 1 1. Find the equation of a hyperbola with a focal points at (4, 0) and (-4,0) and vertices at (3,0) and (-3, 0). 13. Some lamps cast light on the wall in the shape of a hyperbola. Determine the equation of that describes the outline of the light. M. Winking Unit 6-4 page 118

22 1. Sec 6.5 Conic Sections Systems Name: Find all of the solutions to the following conic systems of equations by graphing and verify the solutions. 1. x + y = 5. x + y = 5 3. x 64 + y 1 = 1 x + 5 = y (x 3) 9 + y 16 = 1 x 4 y 3 = 1 Find the solutions to the nearest hundredth of the systems by graphing with your calculator and sketch a graph 4. y = x and x + y = 9 M. Winking Unit 6-5 pg. 119

23 Find the solutions to the nearest hundredth of the systems by graphing with your calculator and sketch a graph 5. x + (y 1) 4 5 = 1 and y x 16 4 = 1 Find the exact solutions to the systems of conic equations by substitution. 6. y = x + 1 and x + y = x + y = 5 and 4x + 36y = 144 M. Winking Unit 6-5 pg. 10

24 Find the exact solutions to the systems of conic equations by substitution. 8. x + 1 = y and x + y = 7 Find the exact solutions to the systems of conic equations by elimination. 9. 4x + 9y = 36 4x 4y = x + y = 5 x + y = 5 M. Winking Unit 6-5 pg. 11

25 Find the exact solutions to the systems of conic equations by elimination x 8y = 48 6y x = 1 1. The comet 41P and Mars Orbits come in very close proximity at times. Luckily, their orbits are not in the same plane as shown at the right. The two planes in which each other orbit just miss one another. However, if we look at a top down or orthogonal view of the two orbits we can find where their orthogonal projections intersect and when they could potentially be at one of their closest points to each other if Mars and 41P s timing is just right. Consider laying out a coordinate grid that goes through the plane in which the orbit of Mars exists. Let s use Astronomical Units (AU) for measurements which represents the mean distance from the Earth to the Sun or roughly 150 million Km Let the Sun be located at roughly (, 0) which would be a focal point of the comet s orbital projection 41P s orbit could be roughly described by 5. 4x y = Mars orbit could be roughly described by (x ) + y =. 3 At what coordinate points would the two orbits potentially almost intersect? 13. Draw a sketch of graph where an ellipse and a hyperbola intersect the requested number of times or put not possible. a. 0 intersections b. 1 intersection c. intersections d. 3 intersections e. 4 intersections M. Winking Unit 6-5 pg. 1