Conics Sequences And Series Probability

83 Parabolas Student Outcomes Students learn the vertex form of the equation of a parabola and how is arises from the definition of a parabola Students perform geometric operations such as rotations reflections and translations on arbitrary parabolas to discover standard representations for their congruence classes In doing so they learn that all parabolas with the same distance p between the focus and the directrix are congruent to the graph of y 12px2 Exercises 1 Show that if the point with coordinates 𝑥 𝑦 is equidistant from 43 and the line 𝑦 5 then 𝑦 1 4 𝑥2 2𝑥 2 Show that if the point with coordinates 𝑥 𝑦 is equidistant from the point 20 and the line 𝑦 4 then 𝑦 1 8 𝑥 22 2 3 Find the equation of the set of points which are equidistant from 02 and the 𝑥axis Sketch this set of points 4 Find the equation of the set of points which are equidistant from the origin and the line 𝑦 6 Sketch this set of points 5 Find the equation of the set of points which are equidistant from 4 2 and the line 𝑦 4 Sketch this set of points 6 Find the equation of the set of points which are equidistant from 40 and the 𝑦axis Sketch this set of points 7 Find the equation of the set of points which are equidistant from the origin and the line 𝑥 2 Sketch this set of points 8 Use the definition of a parabola to sketch the parabola defined by the given focus and directrix a Focus 05 Directrix 𝑦 1 b Focus 20 Directrix 𝑦axis c Focus 4 4 Directrix 𝑥axis d Focus 24 Directrix 𝑦 2 9 Find an analytic equation for each parabola described in Problem 8 10 Are any of the parabolas described in Problem 9 congruent Explain your reasoning 11 Sketch each parabola labeling its focus and directrix a 𝑦 1 2 𝑥2 2 b 𝑦 1 4 𝑥2 1 c 𝑥 1 8 𝑦2 d 𝑥 1 2 𝑦2 2 e 𝑦 1 10 𝑥 12 2 12 Determine which parabolas are congruent to the parabola with equation 𝑦 1 4 𝑥2 a c b d 13 Determine which equations represent the graph of a parabola that is congruent to the parabola shown to right a 𝑦 1 20 𝑥2 b 𝑦 1 10 𝑥2 3 c 𝑦 1 20 𝑥2 8 d 𝑦 1 5 𝑥2 5 e 𝑥 1 10 𝑦2 f 𝑥 1 5 𝑦 32 g 𝑥 1 20 𝑦2 1 14 Jemma thinks that the parabola with equation 𝑦 1 3 𝑥2 is NOT congruent to the parabola with equation 𝑦 1 3 𝑥2 1 Do you agree or disagree Create a convincing argument to support your reasoning 15 Let 𝑃 be the parabola with focus 26 and directrix 𝑦 2 a Write an equation whose graph is a parabola congruent to 𝑃 with focus 04 b Write an equation whose graph is a parabola congruent to 𝑃 with focus 00 c Write an equation whose graph is a parabola congruent to 𝑃 with the same directrix but different focus d Write an equation whose graph is a parabola congruent to 𝑃 with the same focus but with a vertical directrix 16 Let 𝑃 be the parabola with focus 04 and directrix 𝑦 𝑥 a Sketch this parabola b By how many degrees would you have to rotate 𝑃 about the focus to make the directrix line horizontal c Write an equation in the form 𝑦 1 2𝑎 𝑥2 whose graph is a parabola that is congruent to 𝑃 d Write an equation whose graph is a parabola with a vertical directrix that is congruent to 𝑃 e Write an equation whose graph is 𝑃 the parabola congruent to 𝑃 that results after 𝑃 is rotated clockwise 45 about the focus f Write an equation whose graph is 𝑃 the parabola congruent to 𝑃 that results after the directrix of 𝑃 is rotated 45 about the origin Extension 17 Consider the function 𝑓𝑥 2𝑥28𝑥9 𝑥24𝑥5 where 𝑥 is a real number a Use polynomial division to rewrite 𝑓 in the form 𝑓𝑥 𝑞 𝑟 𝑥24𝑥5 for some real numbers 𝑞 and 𝑟 b Find the 𝑥value where the maximum occurs for the function 𝑓 without using graphing technology Explain how you know 85 Ellipses Student Outcomes Students derive the equations of ellipses given the foci using the fact that the sum of distances from the foci is constant Exercises 1 Derive the equation of the ellipse with the given foci 𝐹 and 𝐺 that passes through point 𝑃 Write your answer in standard form 𝑥2 𝑎2 𝑦2 𝑏2 1 a The foci are 𝐹20 and 𝐺20 and point 𝑃𝑥 𝑦 satisfies the condition 𝑃𝐹 𝑃𝐺 5 b The foci are 𝐹10 and 𝐺10 and point 𝑃𝑥 𝑦 satisfies the condition 𝑃𝐹 𝑃𝐺 5 c The foci are 𝐹0 1 and 𝐺01 and point 𝑃𝑥 𝑦 satisfies the condition 𝑃𝐹 𝑃𝐺 4 d The foci are 𝐹 2 3 0 and 𝐺 2 3 0 and point 𝑃𝑥 𝑦 satisfies the condition 𝑃𝐹 𝑃𝐺 3 e The foci are 𝐹0 5 and 𝐺05 and point 𝑃𝑥 𝑦 satisfies the condition 𝑃𝐹 𝑃𝐺 12 f The foci are 𝐹60 and 𝐺60 and point 𝑃𝑥 𝑦 satisfies the condition 𝑃𝐹 𝑃𝐺 20 1 Recall from Lesson 6 that the semimajor axes of an ellipse are the segments from the center to the farthest vertices and the semiminor axes are the segments from the center to the closest vertices For each of the ellipses in Problem 1 find the lengths 𝑎 and 𝑏 of the semimajor axes 2 Summarize what you know about equations of ellipses centered at the origin with vertices 𝑎 0 𝑎 0 0 𝑏 and 0 𝑏 3 Use your answer to Problem 3 to find the equation of the ellipse for each of the situations below a An ellipse centered at the origin with 𝑥intercepts 20 20 and 𝑦 intercepts 08 0 8 b An ellipse centered at the origin with 𝑥intercepts 5 0 5 0 and 𝑦 intercepts 03 0 3 4 Examine the ellipses and the equations of the ellipses you have worked with and describe the ellipses with equation 𝑥2 𝑎2 𝑦2 𝑏2 1 in the three cases 𝑎 𝑏 𝑎 𝑏 and 𝑏 𝑎 5 Is it possible for 𝑥2 4 𝑦2 9 1 to have foci at 𝑐 0 and 𝑐 0 for some real number 𝑐 6 For each value of 𝑘 specified in parts ae plot the set of points in the plane that satisfy the equation 𝑥2 4 𝑦2 𝑘 a 𝑘 1 b 𝑘 1 4 c 𝑘 1 9 d 𝑘 1 16 e 𝑘 1 25 f 𝑘 1 100 g Make a conjecture Which points in the plane satisfy the equation 𝑥2 4 𝑦2 0 h Explain why your conjecture in part g makes sense algebraically i Which points in the plane satisfy the equation 𝑥2 4 𝑦2 1 7 For each value of 𝑘 specified in parts ae plot the set of points in the plane that satisfy the equation 𝑥2 𝑘 𝑦2 1 a 𝑘 1 b 𝑘 2 c 𝑘 4 d 𝑘 15 e 𝑘 30 f Describe what happens to the graph of 𝑥2 𝑘 𝑦2 1 as 𝑘 8 For each value of 𝑘 specified in parts ae plot the set of points in the plane that satisfy the equation 𝑥2 𝑦2 𝑘 1 a 𝑘 1 b 𝑘 2 c 𝑘 4 d 𝑘 15 e 𝑘 30 f Describe what happens to the graph of 𝑥2 𝑦2 𝑘 1 as 𝑘 2 a 12 2642 12 2 4 6 2 12 6 8 12 6 12 8 3 4 b 12 3751 12 3 5 7 1 12 8 8 12 8 12 8 4 4 c 12 8 10 9 5 12 8 9 10 5 12 1 5 12 1 52 d 12 3 2 6 9 12 3 6 2 9 12 3 7 32 72 3 3 a x 02 y 02 152 x2 y2 225 b x 02 y 02 7 sqrtr2 72 27 sqrtr r2 x2 y2 49 14 sqrtr r2 c x32 y52 92 x 32 y 52 81 d x 22 y 52 r2 x 22 y 52 49 e Theres r R such that 3 32 2 62 r2 r sqrt3 32 2 62 sqrt02 82 sqrt64 8 x 32 y 62 82 x 32 y 62 64 7 81 1 a d5114 sqrt5 12 1 42 sqrt5 12 32 sqrt42 9 sqrt16 9 sqrt25 5 b d2545 sqrt2 42 5 52 sqrt32 02 sqrt9 3 c d8273 sqrt8 72 2 32 sqrt8 72 2 32 sqrt152 52 sqrt225 25 sqrt250 158 d d1455 sqrt5 12 5 42 sqrt42 12 sqrt16 1 sqrt17 41 f There is r R such that 422 422 r2 r 422 422 22 22 44 8 x22 y22 82 x22 y22 8 4 i 2x2 2y2 450 12 2x2 2y2 12 450 x2 y2 225 152 standard form Therefore the center is 0 0 and the radius is 15 ii 3x2 3y2 432 13 3x2 3y2 13 432 x2 y2 144 122 standard form Therefore the center of the circle is 00 and its radius is 12 iii x32 y52 81 x32 y52 92 standard form Therefore the center of the circle is 3 5 and its radius is 9 x0 32 y52 81 y2 10y 47 0 y 5 62 y0 x32 52 81 x2 6x 47 0 x 3 214 iv x22 y52 49 x22 y52 r2 standard form Therefore the center of the circle is 2 5 and its radius is 7 x 0 22 y52 49 y2 10y 20 0 y 5 85 y0 x22 52 49 x2 4x 20 0 x 2 26 dxy 43 4x² 3y² dxy line y5 y5 xy is equidistant from 43 and y5 dxy 43 dxy line y5 4x² 3y² y5 4x² 3y² y5² y5² x² 8x 4y 4y x² 8x y 14 x² 8x 14 x² 2x dxy 20 2x² 0y² 2x² y² dxy y4 y4 y4 dxy 20 dxy y4 2x² y² y4 2x² y² y4² y4² 2x² y² y² 8y 16 8y 2x² 16 x2² 16 y 18 x2² 16 18 x2² 168 18 x2² 2 dxy 02 0x² 2y² x² y2² dxy y0 y0 y dxy 02 dxy y0 x² y2² y x² y2² y² y² x² 4y 4 0 4y x² 4 y 14 x² 1 4 dxy00 sqrtx2 y2 dxy y6 y 6 dxy00 dxy y6 sqrtx2 y2 y 6 x2 y2 y 62 y 62 y2 12y 36 x2 12y 36 y 112 x2 3612 112 x2 3 y 112 x2 3 5 dxy42 sqrtx 42 y 22 dxy y4 y 4 dxy42 dxy y4 sqrtx42 y22 y 4 x 42 y 22 y 42 y 42 x2 8x 16 y2 4y 4 y2 8y 16 x2 8x 4 12y 0 y 112 x2 23 x 13 6 dxy 40 sqrtx 42 y2 dxy x0 x dxy40 dxy x0 sqrtx42 y2 x x 42 y2 x2 x2 x2 8x 16 y2 x2 x 18 y2 2 x 18 y2 2 dxy00x2y2 dxyx2x2 dxy00dxyx2 x2y2x2 x2y2x22x22 x2y2x24x4 x14 y2 1 x14 y2 1 8 a 05 y 1 b 20 c 4 4 d 4 2 y 2 9 a x2 y52 y12 x2 y2 10 y 25 y2 2 y 1 y112 x2 2 b x22 y2 x2 x2 4x 4 y2 x2 x 14 y2 1 c x42 y42 y2 x2 8x 16 y2 8y 16 y2 y 18 x2 x 2 d x22 y42 y22 x2 4x y2 8y 16 y2 4x 4 y 18 x2 x 2 10 Yes c and d are congruent parabolas The difference between them is one is the negative of the other ie c opens downwards and d opens upwards but they have the same size and shape because of the modulus of their coefficients being the same 11 d 2y x2 4 Focus 1 52 Directrix y 32 b y 14 x2 1 Focus 9 0 Directrix y 2 10 c x 18 y2 Focus 2 0 Directrix x 2 d x 12 y2 2 Focus 52 0 Directrix x 32 e y 110 x12 2 Focus 1 12 Directrix y 92 12 11 87 Hyperbolas Exercises 1 In the following exercises graph 2 In the following exercises graph 3 In the following exercises write the equation in standard form and graph 93 Binomial Theorem Student Outcomes Students observe patterns in the coefficients of the terms in binomial expansions They formalize their observations and explore the mathematical basis for them Students use the binomial theorem to solve problems in a geometric context Exercises 1 Consider the binomial 2𝑢 3𝑣6 a Find the term that contains 𝑣4 b Find the term that contains 𝑢3 c Find the third term 2 Consider the binomial 𝑢2 𝑣36 a Find the term that contains 𝑣6 b Find the term that contains 𝑢6 c Find the fifth term 3 Find the sum of all coefficients in the following binomial expansion a 2𝑢 𝑣10 b 2𝑢 𝑣10 c 2𝑢 3𝑣11 d 𝑢 3𝑣11 e 1 𝑖10 f 1 𝑖10 g 1 𝑖200 h 1 𝑖201 4 Expand the binomial 1 2𝑖 6 5 Show that 2 2𝑖 20 2 2𝑖 20 is an integer 6 We know 𝑢 𝑣2 𝑢2 2𝑢𝑣 𝑣2 𝑢2 𝑣2 2𝑢𝑣 Use this pattern to predict what the expanded form of each expression would be Then expand the expression and compare your results a 𝑢 𝑣 𝑤2 b 𝑎 𝑏 𝑐 𝑑2 7 Look at the powers of 101 up to the fourth power on a calculator Explain what you see Predict the value of 1015 and then find the answer on a calculator Are they the same 8 Can Pascals triangle be applied to 1 𝑢 1 𝑣 𝑛 given 𝑢 𝑣 0 9 The volume and surface area of a sphere are given by 𝑉 4 3 𝜋𝑟3 and 𝑆 4𝜋𝑟2 Suppose we increase the radius of a sphere by 0001 units from 𝑟 to 𝑟 0001 a Use the binomial theorem to write an expression for the increase in volume 𝑉𝑟 0001 𝑉𝑟 as the sum of three terms b Write an expression for the average rate of change of the volume as the radius increases from 𝑟 to 𝑟 0001 c Simplify the expression in part b to compute the average rate of change of the volume of a sphere as the radius increases from 𝑟 to 𝑟 0001 d What does the expression from part c resemble e Why does it make sense that the average rate of change should approximate the surface area Think about the geometric figure formed by 𝑉𝑟 0001 𝑉𝑟 What does this represent f How could we approximate the volume of the shell using surface area And the average rate of change for the volume g Find the difference between the average rate of change of the volume and 𝑆𝑟 when 𝑟 1 10 The area and circumference of a circle of radius 𝑟 are given by 𝐴𝑟 𝜋𝑟2 and 𝐶𝑟 2𝜋𝑟 Suppose we increase the radius of a sphere by 0001 units from 𝑟 to 𝑟 0001 a Use the binomial theorem to write an expression for the increase in area volume 𝐴𝑟 0001 𝐴𝑟 as a sum of three terms b Write an expression for the average rate of change of the area as the radius increases from 𝑟 to 𝑟 0001 c Simplify the expression in part b to compute the average rate of change of the area of a circle as the radius increases from 𝑟 to 𝑟 0001 d What does the expression from part c resemble e Why does it make sense that the average rate of change should approximate the area of a circle Think about the geometric figure formed by 𝐴𝑟 0001 𝐴𝑟 What does this represent f How could we approximate the area of the shell using circumference And the average rate of change for the area g Find the difference between the average rate of change of the area and 𝐶𝑟 when 𝑟 1 91 Sequences Student Outcomes Students recognize when a table of values represents an arithmetic or geometric sequence Patterns are present in tables of values They choose and define the parameter values for a function that represents a sequence Summary A sequence is a list of numbers or objects in a special order An arithmetic sequence goes from one term to the next by adding or subtracting the same value A geometric sequence goes from one term to the next by multiplying or dividing by the same value Looking at the difference of differences can be a quick way to determine if a sequence can be represented as a quadratic expression Exercises Solve the following problems by finding the functionformula that represents the 𝑛th term of the sequence 1 After a knee injury a jogger is told he can jog 10 minutes every day and that he can increase his jogging time by 3 minutes every two weeks How long will it take for him to be able to jog one hour a day 2 A ball is dropped from a height of 15 feet The ball then bounces to 80 of its previous height with each subsequent bounce a Explain how this situation can be modeled with a sequence Week Daily Jog Time 1 10 2 10 3 13 4 13 5 16 6 16 b How high to the nearest tenth of a foot does the ball bounce on the fifth bounce 3 Consider the following sequence 8 17 32 53 80 113 a What pattern do you see and what does that pattern mean for the analytical representation of the function b What is the symbolic representation of the sequence 4 Arnold wants to be able to complete 100 militarystyle pull ups His trainer puts him on a workout regimen designed to improve his pullup strength The following chart shows how many pullups Arnold can complete after each month of training How many months will it take Arnold to achieve his goal if this pattern continues Month PullUp Count 1 2 2 5 3 10 4 17 5 26 6 37 103 Probability events Student Outcomes Students represent events by shading appropriate regions in a Venn diagram Given a chance experiment with equally likely outcomes students calculate counts and probabilities by addingsubtracting given counts or probabilities Students interpret probabilities in context Sumary In a probability experiment the events can be represented by circles in a Venn diagram Combinations of events using and or and not can be shown by shading the appropriate regions of the Venn diagram The number of possible outcomes can be shown in each region of the Venn diagram alternatively probabilities may be shown The number of outcomes in a given region or the probability associated with it can be calculated by adding or subtracting the known numbers of possible outcomes or probabilities Exercises 1 On a flight some of the passengers have frequentflier status and some do not Also some of the passengers have checked baggage and some do not Let the set of passengers who have frequentflier status be 𝐹 and the set of passengers who have checked baggage be 𝐶 On the Venn diagrams provided shade the regions representing the following instances 1 Passengers who have frequentflier status and have checked baggage 2 Passengers who have frequentflier status or have checked baggage 3 Passengers who do not have both frequentflier status and checked baggage 4 Passengers who have frequentflier status or do not have checked baggage 2 For the scenario introduced in Problem 1 suppose that of the 400 people on the flight 368 have checked baggage 228 have checked baggage but do not have frequentflier status and 8 have neither frequentflier status nor checked baggage 1 Using a Venn diagram calculate the following 1 The number of people on the flight who have frequentflier status and have checked baggage 2 The number of people on the flight who have frequentflier status 2 In the Venn diagram provided below write the probabilities of the events associated with the regions marked with a star 3 When an animal is selected at random from those at a zoo the probability that it is North American meaning that its natural habitat is in the North American continent is 065 the probability that it is both North American and a carnivore is 016 and the probability that it is neither American nor a carnivore is 017 1 Using a Venn diagram calculate the probability that a randomly selected animal is a carnivore 2 Complete the table below showing the probabilities of the events corresponding to the cells of the table North American Not North American Total Carnivore Not Carnivore Total 4 This question introduces the mathematical symbols for and or and not Considering all the people in the world let 𝐴 be the set of Americans citizens of the United States and let 𝐵 be the set of people who have brothers The set of people who are Americans and have brothers is represented by the shaded region in the Venn diagram below This set is written 𝐴 𝐵 read 𝐴 intersect 𝐵 and the probability that a randomly selected person is American and has a brother is written 𝑃𝐴 𝐵 The set of people who are Americans or have brothers is represented by the shaded region in the Venn diagram below This set is written 𝐴 𝐵 read 𝐴 union 𝐵 and the probability that a randomly selected person is American or has a brother is written 𝑃𝐴 𝐵 The set of people who are not Americans is represented by the shaded region in the Venn diagram below This set is written 𝐴𝐶 read 𝐴 complement and the probability that a randomly selected person is not American is written 𝑃𝐴𝐶 Now think about the cars available at a dealership Suppose a car is selected at random from the cars at this dealership Let the event that the car has manual transmission be denoted by 𝑀 and let the event that the car is a sedan be denoted by 𝑆 The Venn diagram below shows the probabilities associated with four of the regions of the diagram 1 What is the value of 𝑃𝑀 𝑆 2 Complete this sentence using and or or 𝑃𝑀 𝑆 is the probability that a randomly selected car has a manual transmission is a sedan 3 What is the value of 𝑃𝑀 𝑆 4 Complete this sentence using and or or 𝑃𝑀 𝑆 is the probability that a randomly selected car has a manual transmission is a sedan 5 What is the value of 𝑃𝑆𝐶 6 Explain the meaning of 𝑃𝑆𝐶 101 Terminology Student Outcomes Students determine the sample space for a chance experiment Given a description of a chance experiment and an event students identify the subset of outcomes from the sample space corresponding to the complement of an event Given a description of a chance experiment and two events students identify the subset of outcomes from the sample space corresponding to the union or intersection of two events Students calculate the probability of events defined in terms of unions intersections and complements for a simple chance experiment with equally likely outcomes Sumary Sample Space The sample space of a chance experiment is the collection of all possible outcomes for the experiment Event An event is a collection of outcomes of a chance experiment For a chance experiment in which outcomes of the sample space are equally likely the probability of an event is the number of outcomes in the event divided by the number of outcomes in the sample space Some events are described in terms of or and or not Exercises Consider a second scenario card that Alan created for his game Scenario Card 2 Tools Spinner 1 Spinner 2 a spinner with six equal sectors Place the number 1 in a sector the number 2 in a second sector the number 3 in a third sector the number 4 in a fourth sector the number 5 in a fifth sector and the number 6 in the last sector Directions chance experiment Spin Spinner 1 and spin Spinner 2 Record the number from Spinner 1 and record the number from Spinner 2 Five Events of Interest Outcome is an odd number on Spinner 2 Outcome is an odd number on Spinner 1 and an even number on Spinner 2 Outcome is the sum of 7 from the numbers received from Spinner 1 and Spinner 2 Outcome is an even number on Spinner 2 Outcome is the sum of 2 from the numbers received from Spinner 1 and Spinner 2 Player Scoring Card for Scenario 2 Turn Outcome from Spinner 1 Outcome from Spinner 2 Points 1 2 3 4 5 1 Prepare Spinner 1 and Spinner 2 for the chance experiment described on this second scenario card Recall that Spinner 2 has six equal sectors 2 What is the sample space for the chance experiment described on this scenario card 3 Based on the sample space determine the outcomes and the probabilities for each of the events on this scenario card Complete the table below Event Outcomes Probability Outcome is an odd number on Spinner 2 Outcome is an odd number on Spinner 1 and an even number on Spinner 2 Outcome is the sum of 7 from the numbers received from Spinner 1 and Spinner 2 Outcome is an even number on Spinner 2 Outcome is the sum of 2 from the numbers received from Spinner 1 and Spinner 2 4 Assign the numbers 15 to the events described on the scenario card Five Events of Interest Scenario 2 Outcome is an odd number on Spinner 2 Outcome is an odd number on Spinner 1 and an even number on Spinner 2 Outcome is the sum of 7 from the numbers received from Spinner 1 and Spinner 2 Outcome is an even number on Spinner 2 Outcome is the sum of 2 from the numbers received from Spinner 1 and Spinner 2 5 Determine at least three final scores based on the numbers you assigned to the events Player Scott Trial Outcome from Spinner 1 Outcome from Spinner 2 Points see Problem 4 1 2 3 4 5 Player Scott Trial Outcome from Spinner 1 Outcome from Spinner 2 Points see Problem 4 1 2 3 4 5 Player Scott Trial Outcome from Spinner 1 Outcome from Spinner 2 Points see Problem 4 1 2 3 4 5 6 Alan also included a fair coin as one of the scenario tools Develop a scenario card Scenario Card 3 that uses the coin and one of the spinners Include a description of the chance experiment and descriptions of five events relevant to the chance experiment Scenario Card 3 Tools Fair coin head or tail Spinner 1 Directions chance experiment Five Events of Interest 7 Determine the sample space for your chance experiment Then complete the table below for the five events on your scenario card Assign the numbers 15 to the descriptions you created Event Outcomes Probability 8 Determine a final score for your game based on five turns Turn Points 1 2 3 4 5 81 Circles Exercises 1 In the following exercises find the distance between the points Round to the nearest tenth if needed a 51 and 14 b 25 and 15 c 82 and 7 3 d 1 4 and 5 5 2 In the following exercises find the midpoint of the line segments whose endpoints are given a 2 6 and 4 2 b 37 and 51 c 8 10 and 95 d 32 and 6 9 3 In the following exercises write the standard form of the equation of the circle with the given information a Radius is 15 and center is 00 b Radius is 7 and center is 00 c Radius is 9 and center is 35 d Radius is 7 and center is 2 5 e Center is 36 and a point on the circle is 3 2 f Center is 22 and a point on the circle is 44 4 In the following exercises 41 Find the center and radius then graph each circle i2x22y2450 ii3x23y2432 iiix32y5281 ivx22y5249 89 Systems of Nonlinear Equations Exercises 1 In the following exercises solve the system of equations by using graphing 2 In the following exercises solve the system of equations by using substitution 3 In the following exercises solve the system of equations by using elimination 4 In the following exercises solve the problem using a system of equations a The sum of the squares of two numbers is 25 The difference of the numbers is 1 Find the numbers b The difference of the squares of two numbers is 45 The difference between the square of the first number and twice the square of the second number is 9 Find the numbers c The perimeter of a rectangle is 58 meters and its area is 210 square meters Find the length and width of the rectangle B5 1 a PF PG 5 2a 5 a 52 c 12 d2 0 2 0 12 sqrt42 2 a2 b2 c2 522 b2 22 b sqrt254 4 sqrt254 164 sqrt94 32 x2522 y2322 1 b PF PG 5 a 52 c 12 d1 0 1 0 1 b sqrt254 1 sqrt254 44 sqrt214 sqrt212 x2522 y2sqrt2122 1 c PF PG 4 a 2 b sqrt22 12 sqrt3 x222 y2sqrt32 1 d PF PG 3 a 32 c 32 b sqrt322 232 sqrt94 49 sqrt811636 sqrt656 x2322 y2sqrt6562 1 e PF PG 12 a 6 c 5 b sqrt62 52 sqrt11 x262 y2sqrt112 1 f PF PG 20 a 10 c 6 b sqrt102 62 8 x2102 y282 1 d a 52 b 32 b a 52 b sqrt212 c a 2 b sqrt3 d a 32 b sqrt656 e a 6 b sqrt11 f a 10 b 8 2 x2a2 y2b2 1 if a b the major axes lies on the xaxis If b a the major axes lies on the yaxis 3 a x2 4 y2 64 1 b x2 5 y2 9 1 4 a b the major axis lies on the xaxis a b thats a circle with radius 0 b a the major axis lies on the yaxis 5 Yes 9 4 c2 c sqrt94 sqrt5 6 a b x2 4 14 y2 4 1 x2 1 y2 122 1 c x2 4 19 y2 9 1 x2 232 y2 132 1 d x2 4 116 y2 116 1 x2 122 y2 142 1 e x2 4 125 y2 125 1 x2 252 y2 152 1 f x2 2102 y2 1102 1 g x2 4 y2 0 y2 x2 4 x2 4 y2 0 x y 0 h For all k in R1 k2 0 y2 0 x2 4 0 x2 4 0 y2 x2 4 0 x 0 y i x24 y2 1 There is no point in R2 satisfying x24 y2 1 x0 0 0 0 7 7 2 a x2 y2 1 circumference b x222 y2 1 c x222 y2 1 7 d x2152 y2 1 e x2302 y2 1 f As k increases the axes lying on x increase either The values of y are independent to k1 so the axes on the yaxis stays the same length a x2 y2 1 circumference 8 8 b x2 y222 1 c x2 y222 1 d x2 y2152 1 9 e x2 y2 302 1 A The values of x are independent of k so the ones lying on the xaxis stay the same length as the ones lying on the yaxis go to infinity and the graph approaches the lines x 1 and x 1