The Ellipse. is larger than the other.

Size: px
Start display at page:

Download "The Ellipse. is larger than the other."

Transcription

1 The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion) in the shpe of ellipse. Ellipses re ommonl noted s the tul pths the plnets tke when oriting the sun. Its formul is derived from the eqution of irle, r, where one of the oeffiients of or is lrger thn the other. The Ellipse in Terms of Lous of Points Version An Ellipse is the set of ll points suh tht the sum of the distnes from two fied points, lled foi, is onstnt. F F The two distnes from point on the ellipse to its foi re the fol rdii.

2 The esiest nd most ville method for drwing n ellipse is with two nils, penil, nd piee of string. We egin tring the string t the two ends to mke loop. Net we put the push nils into wll or desk (the nils re the foi) nd l the string round the nils. Lstl we put the penil inside the string nd pull it tight round the nils. The tre tht we produe will e n ellipse (visit first We Link for nimtion). The Ellipse in Terms of Lous of Points Version The trditionl w to define n ellipse, s oni setion. It is the set of points whose distne from point (the fous) nd line (the diretri) is in onstnt proportion).

3 The Antom of n Ellipse B V' F' O F V P B' The point O, the midpoint F F, is the entre of the ellipse. Eh of the foi is units from the entre. The points V nd V re the verties of the ellipse. The verties re where the is joining the foi (Fol Ais) interepts the ellipse. The line segment V V is the mjor is. The mjor is lies on top of the Fol Ais nd is n is of smmetr. Sine eh verte is units from the entre, therefore the length of the mjor is is. The points B B re the endpoints of the minor is. The minor is is the line segment through the entre of the ellipse perpendiulr to the mjor is. The length of the minor is is. Note: the minor is is lws shorter thn the mjor is. The sum of the distne from the two foi to n point on the ellipse is The distnes,, nd re relted the eqution. Some ellipses re more elongted tht other ellipses. The loser the foi re to the entre the more the ellipse eomes irulr. We ll the rtio of the eentriit (e). The Eentriit Beuse e nd 0, then 0 e. The loser e is to 0, the loser the foi re to the entre nd the more irulr the ellipse. The loser e is to, the loser the foi re to the verties nd the more elongted the ellipse.

4 Deriving n horizontl Ellipse Eqution with Centre t (0, 0) nd foi Let P(,) represent n point on the ellipse, nd let F nd F represent the two foi of the ellipse. Then PF + PF = Now rewriting PF + PF = in terms of distnes >,0 If we ple rdil on either side it is esier to squre nd remove one of the rdil epressions. Agin get the rdil itself so squring will eliminte rdil sign. Similrl the Vertil Ellipse eqution is with foi t 0,, nd > The stndrd form of the equtions for the ellipse whose entre is (h,k) is h k, for horizontl mjor is, where >, nd foi h, k h k, for vertil mjor is, where >, nd foi h, k

5 Emple Desrie the ke properties of 5 6 Centre nd Orienttion of the Ellipse The ellipse is in the form entre (0, 0). with >, then this is n Horizontl ellipse with Lengths of Mjor nd Minor Ais Sine 5 nd 6, we hve 5 nd 4. Therefore the length of the Mjor is is, nd the Minor Ais is interepts To find the -interepts (verties, sine origin is (0, 0)), set =0 nd solve for. so 5 5 Hene the -interepts re 5,0 nd 5,0 -interepts To find the -interepts set =0 nd solve for., so 4 6 Hene the -interepts re nd 0,4 0, 4 Horizontl Ellipse Lotion of Foi Sine 5 6 9, =3. The foi re on the -is, 3 units on either side of the origin t 3,0 nd 3,0

6 Emple Find the eqution of the ellipse with entre (0,0), foi (5,0) nd (-5,0) nd -interepts 6 nd 6. The foi re on the -is. Therefore, the mjor is is horizontl. The eqution, where >, nd =6 nd =5. Find interepts re 6 nd -6 Foi t 5 nd -5 Therefore 6 36 Emple Find the entre nd the length of the mjor nd the minor es for the ellipse with the following eqution: Chnge into stndrd form dividing The entre of the ellipse is (,-), nd =4 nd = The length of the mjor is is =(4)=8, nd the length of the minor is is =()=4. Emple width The shpe of n rh is semi-ellipse, nd its spn is the mjor is of the ellipse. If the spn is 60m nd its height is 0m, find the height of the rh t point m from the entre, to the nerest tenth of metre.

7 With these tpe of rhw question, it is est to mke the entre of the rod the origin. Then we hve hlf the rod of the left nd the other hlf on the right of the entre (0, 0) with the tllest height ove the middle of the rod. Letting the point (, ) e on the ellipse, then we need onl determine. Sine this is horizontl ellipse, we hve with =30, =0 Thus the eqution of the ellipse is Sustitute = in the eqution, nd solve for Sine must e positive, the height of the rh is pproimtel 8.3m

8 Emple Determine the verties nd foi for the following ellipse 3 8 Rewrite eqution of ellipse in the form, dividing 8. Vertil Ellipse Therefore 6 nd 3. Sine 9>6 we hve vertil ellipse with verte t (0, 3) nd (0, -3) Now for n ellipse we know The foi re: 0, 3, 0, 3 Emple 3 Determine the verties nd foi for the following ellipse This is horizontl ellipse with entre h, k,3 Also Foi re: h, k 8,3 9,3, 7,3 Verties re: h, k 0,3,3, 9,

9 Emple Determine the eqution of the ellipse with mjor is 3, entre,7.,, nd one fous t We know h, k, nd tht sine the -vlue of oth the entre nd the fous re the sme (), then this is vertil ellipse. vlue of verte We lso know tht =3, or =6 nd tht Now h, k so 7 9 Onl vlue of possile The generl eqution is, therefore Emple The verties of n ellipse with eentriit 0.8 re loted t the stndrd form of the eqution of the ellipse., nd 8,. Determine First determine if the ellipse is horizontl or vertil. Sine the verties hve the sme - oordinte ut different -oordintes, the verties lie on horizontl line, so this is horizontl ellipse. The entre (h, k) of the ellipse is on the line tht ontins the verties nd is midw etween them. 8 Therefore h, k, 3, Let s find : The distne etween (3, ) nd (8, ) is 5 units thus 5 nd 5 Find : sine e

10 Now for : h k Putting it ll together : 3 5 9

Ellipses. The second type of conic is called an ellipse.

Ellipses. The second type of conic is called an ellipse. Ellipses The seond type of oni is lled n ellipse. Definition of Ellipse An ellipse is the set of ll points (, y) in plne, the sum of whose distnes from two distint fied points (foi) is onstnt. (, y) d

More information

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a. Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining

More information

8.3 THE HYPERBOLA OBJECTIVES

8.3 THE HYPERBOLA OBJECTIVES 8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS

More information

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr

More information

10.2 The Ellipse and the Hyperbola

10.2 The Ellipse and the Hyperbola CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point

More information

8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1

8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1 8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.

More information

Comparing the Pre-image and Image of a Dilation

Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

More information

Algebra II Notes Unit Ten: Conic Sections

Algebra II Notes Unit Ten: Conic Sections Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting

More information

m A 1 1 A ! and AC 6

m A 1 1 A ! and AC 6 REVIEW SET A Using sle of m represents units, sketh vetor to represent: NON-CALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in north-esterl diretion. Simplif:

More information

P 1 (x 1, y 1 ) is given by,.

P 1 (x 1, y 1 ) is given by,. MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round

More information

( ) { } [ ] { } [ ) { } ( ] { }

( ) { } [ ] { } [ ) { } ( ] { } Mth 65 Prelulus Review Properties of Inequlities 1. > nd > >. > + > +. > nd > 0 > 4. > nd < 0 < Asolute Vlue, if 0, if < 0 Properties of Asolute Vlue > 0 1. < < > or

More information

THREE DIMENSIONAL GEOMETRY

THREE DIMENSIONAL GEOMETRY MD THREE DIMENSIONAL GEOMETRY CA CB C Coordintes of point in spe There re infinite numer of points in spe We wnt to identif eh nd ever point of spe with the help of three mutull perpendiulr oordintes es

More information

Reflection Property of a Hyperbola

Reflection Property of a Hyperbola Refletion Propert of Hperol Prefe The purpose of this pper is to prove nltill nd to illustrte geometrill the propert of hperol wherein r whih emntes outside the onvit of the hperol, tht is, etween the

More information

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272.

Geometry of the Circle - Chords and Angles. Geometry of the Circle. Chord and Angles. Curriculum Ready ACMMG: 272. Geometry of the irle - hords nd ngles Geometry of the irle hord nd ngles urriulum Redy MMG: 272 www.mthletis.om hords nd ngles HRS N NGLES The irle is si shpe nd so it n e found lmost nywhere. This setion

More information

50 AMC Lectures Problem Book 2 (36) Substitution Method

50 AMC Lectures Problem Book 2 (36) Substitution Method 0 AMC Letures Prolem Book Sustitution Metho PROBLEMS Prolem : Solve for rel : 9 + 99 + 9 = Prolem : Solve for rel : 0 9 8 8 Prolem : Show tht if 8 Prolem : Show tht + + if rel numers,, n stisf + + = Prolem

More information

( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x).

( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x). Mth 15 Fettermn/DeSmet Gustfson/Finl Em Review 1) Let f( ) = 10 5. Find nd simplif f( ) nd then stte the domin of f(). ) Let f( ) = +. Find nd simplif f(1) nd then stte the domin of f(). ) Let f( ) = 8.

More information

12.4 Similarity in Right Triangles

12.4 Similarity in Right Triangles Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right

More information

SIMPLE NONLINEAR GRAPHS

SIMPLE NONLINEAR GRAPHS S i m p l e N o n l i n e r G r p h s SIMPLE NONLINEAR GRAPHS www.mthletis.om.u Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle

More information

8.2: CIRCLES AND ELLIPSES

8.2: CIRCLES AND ELLIPSES 8.: CIRCLES AND ELLIPSES GEOMETRY OF AN ELLIPSE Geometry of n Ellipse Definition: An ellipse is the set of ll points in plne whose distnce from two fixed points in the plne hve constnt sum. Voculry The

More information

Applications of Definite Integral

Applications of Definite Integral Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between

More information

LESSON 11: TRIANGLE FORMULAE

LESSON 11: TRIANGLE FORMULAE . THE SEMIPERIMETER OF TRINGLE LESSON : TRINGLE FORMULE In wht follows, will hve sides, nd, nd these will e opposite ngles, nd respetively. y the tringle inequlity, nd..() So ll of, & re positive rel numers.

More information

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.

More information

Forces on curved surfaces Buoyant force Stability of floating and submerged bodies

Forces on curved surfaces Buoyant force Stability of floating and submerged bodies Stti Surfe ores Stti Surfe ores 8m wter hinge? 4 m ores on plne res ores on urved surfes Buont fore Stbilit of floting nd submerged bodies ores on Plne res Two tpes of problems Horizontl surfes (pressure

More information

Applications of Definite Integral

Applications of Definite Integral Chpter 5 Applitions of Definite Integrl 5.1 Are Between Two Curves In this setion we use integrls to find res of regions tht lie between the grphs of two funtions. Consider the region tht lies between

More information

, are called the foci (plural of focus) of the ellipse. The line segments F 1. P are called focal radii of the ellipse.

, are called the foci (plural of focus) of the ellipse. The line segments F 1. P are called focal radii of the ellipse. 8.5 The Ellipse Kidne stones re crstl-like ojects tht cn form in the kidnes. Trditionll, people hve undergone surger to remove them. In process clled lithotrips, kidne stones cn now e removed without surger.

More information

UNCORRECTED SAMPLE PAGES. Australian curriculum NUMBER AND ALGEBRA

UNCORRECTED SAMPLE PAGES. Australian curriculum NUMBER AND ALGEBRA 7A 7B 7C 7D 7E 7F 7G 7H 7I 7J 7K Chpter Wht ou will lern 7Prols nd other grphs Eploring prols Skething prols with trnsformtions Skething prols using ftoristion Skething ompleting the squre Skething using

More information

Integration. antidifferentiation

Integration. antidifferentiation 9 Integrtion 9A Antidifferentition 9B Integrtion of e, sin ( ) nd os ( ) 9C Integrtion reognition 9D Approimting res enlosed funtions 9E The fundmentl theorem of integrl lulus 9F Signed res 9G Further

More information

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd

More information

Mathematics. Area under Curve.

Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

More information

GM1 Consolidation Worksheet

GM1 Consolidation Worksheet Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up

More information

ES.182A Topic 32 Notes Jeremy Orloff

ES.182A Topic 32 Notes Jeremy Orloff ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In

More information

5. Every rational number have either terminating or repeating (recurring) decimal representation.

5. Every rational number have either terminating or repeating (recurring) decimal representation. CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd

More information

2. There are an infinite number of possible triangles, all similar, with three given angles whose sum is 180.

2. There are an infinite number of possible triangles, all similar, with three given angles whose sum is 180. SECTION 8-1 11 CHAPTER 8 Setion 8 1. There re n infinite numer of possile tringles, ll similr, with three given ngles whose sum is 180. 4. If two ngles α nd β of tringle re known, the third ngle n e found

More information

Exercise sheet 6: Solutions

Exercise sheet 6: Solutions Eerise sheet 6: Solutions Cvet emptor: These re merel etended hints, rther thn omplete solutions. 1. If grph G hs hromti numer k > 1, prove tht its verte set n e prtitioned into two nonempt sets V 1 nd

More information

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL:

PYTHAGORAS THEOREM WHAT S IN CHAPTER 1? IN THIS CHAPTER YOU WILL: PYTHAGORAS THEOREM 1 WHAT S IN CHAPTER 1? 1 01 Squres, squre roots nd surds 1 02 Pythgors theorem 1 03 Finding the hypotenuse 1 04 Finding shorter side 1 05 Mixed prolems 1 06 Testing for right-ngled tringles

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

Introduction. Definition of Hyperbola

Introduction. Definition of Hyperbola Section 10.4 Hperbols 751 10.4 HYPERBOLAS Wht ou should lern Write equtions of hperbols in stndrd form. Find smptotes of nd grph hperbols. Use properties of hperbols to solve rel-life problems. Clssif

More information

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=

I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r= 11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel

More information

SECTION A STUDENT MATERIAL. Part 1. What and Why.?

SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

K 7. Quadratic Equations. 1. Rewrite these polynomials in the form ax 2 + bx + c = 0. Identify the values of a, b and c:

K 7. Quadratic Equations. 1. Rewrite these polynomials in the form ax 2 + bx + c = 0. Identify the values of a, b and c: Qudrti Equtions The Null Ftor Lw Let's sy there re two numers nd. If # = then = or = (or oth re ) This mens tht if the produt of two epressions is zero, then t lest one of the epressions must e equl to

More information

THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

More information

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

More information

Trigonometry Revision Sheet Q5 of Paper 2

Trigonometry Revision Sheet Q5 of Paper 2 Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.

More information

NORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y

NORMALS. a y a y. Therefore, the slope of the normal is. a y1. b x1. b x. a b. x y a b. x y LOCUS 50 Section - 4 NORMALS Consider n ellipse. We need to find the eqution of the norml to this ellipse t given point P on it. In generl, we lso need to find wht condition must e stisfied if m c is to

More information

Logarithms LOGARITHMS.

Logarithms LOGARITHMS. Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the

More information

y z A left-handed system can be rotated to look like the following. z

y z A left-handed system can be rotated to look like the following. z Chpter 2 Crtesin Coördintes The djetive Crtesin bove refers to René Desrtes (1596 1650), who ws the first to oördintise the plne s ordered pirs of rel numbers, whih provided the first sstemti link between

More information

, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF

, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs

More information

GRAND PLAN. Visualizing Quaternions. I: Fundamentals of Quaternions. Andrew J. Hanson. II: Visualizing Quaternion Geometry. III: Quaternion Frames

GRAND PLAN. Visualizing Quaternions. I: Fundamentals of Quaternions. Andrew J. Hanson. II: Visualizing Quaternion Geometry. III: Quaternion Frames Visuliing Quternions Andrew J. Hnson Computer Siene Deprtment Indin Universit Siggrph Tutoril GRAND PLAN I: Fundmentls of Quternions II: Visuliing Quternion Geometr III: Quternion Frmes IV: Clifford Algers

More information

Mathematics Number: Logarithms

Mathematics Number: Logarithms plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement

More information

JEE Advnced Mths Assignment Onl One Correct Answer Tpe. The locus of the orthocenter of the tringle formed the lines (+P) P + P(+P) = 0, (+q) q+q(+q) = 0 nd = 0, where p q, is () hperol prol n ellipse

More information

Ellipse. 1. Defini t ions. FREE Download Study Package from website: 11 of 91CONIC SECTION

Ellipse. 1. Defini t ions. FREE Download Study Package from website:  11 of 91CONIC SECTION FREE Downlod Stud Pckge from wesite: www.tekoclsses.com. Defini t ions Ellipse It is locus of point which moves in such w tht the rtio of its distnce from fied point nd fied line (not psses through fied

More information

UNCORRECTED. Australian curriculum NUMBER AND ALGEBRA

UNCORRECTED. Australian curriculum NUMBER AND ALGEBRA 0A 0B 0C 0D 0E 0F 0G 0H Chpter Wht ou will lern Qudrti equtions (Etending) Solving + = 0 nd = d (Etending) Solving + + = 0 (Etending) Applitions of qudrti equtions (Etending) The prol Skething = with diltions

More information

MATH 115: Review for Chapter 7

MATH 115: Review for Chapter 7 MATH 5: Review for Chpter 7 Cn ou stte the generl form equtions for the circle, prbol, ellipse, nd hperbol? () Stte the stndrd form eqution for the circle. () Stte the stndrd form eqution for the prbol

More information

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2 SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the

More information

Chapter Gauss Quadrature Rule of Integration

Chapter Gauss Quadrature Rule of Integration Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to

More information

3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles.

3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles. 3 ngle Geometry MEP Prtie ook S3 3.1 Mesuring ngles 1. Using protrtor, mesure the mrked ngles. () () (d) (e) (f) 2. Drw ngles with the following sizes. () 22 () 75 120 (d) 90 (e) 153 (f) 45 (g) 180 (h)

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +

More information

Section 4.4. Green s Theorem

Section 4.4. Green s Theorem The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls

More information

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets. I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the

More information

This enables us to also express rational numbers other than natural numbers, for example:

This enables us to also express rational numbers other than natural numbers, for example: Overview Study Mteril Business Mthemtis 05-06 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers

More information

Section 2.3. Matrix Inverses

Section 2.3. Matrix Inverses Mtri lger Mtri nverses Setion.. Mtri nverses hree si opertions on mtries, ition, multiplition, n sutrtion, re nlogues for mtries of the sme opertions for numers. n this setion we introue the mtri nlogue

More information

Individual Group. Individual Events I1 If 4 a = 25 b 1 1. = 10, find the value of.

Individual Group. Individual Events I1 If 4 a = 25 b 1 1. = 10, find the value of. Answers: (000-0 HKMO Het Events) Creted y: Mr. Frnis Hung Lst udted: July 0 00-0 33 3 7 7 5 Individul 6 7 7 3.5 75 9 9 0 36 00-0 Grou 60 36 3 0 5 6 7 7 0 9 3 0 Individul Events I If = 5 = 0, find the vlue

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

Lesson-5 ELLIPSE 2 1 = 0

Lesson-5 ELLIPSE 2 1 = 0 Lesson-5 ELLIPSE. An ellipse is the locus of point which moves in plne such tht its distnce from fied point (known s the focus) is e (< ), times its distnce from fied stright line (known s the directri).

More information

PYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS

PYTHAGORAS THEOREM,TRIGONOMETRY,BEARINGS AND THREE DIMENSIONAL PROBLEMS PYTHGORS THEOREM,TRIGONOMETRY,ERINGS ND THREE DIMENSIONL PROLEMS 1.1 PYTHGORS THEOREM: 1. The Pythgors Theorem sttes tht the squre of the hypotenuse is equl to the sum of the squres of the other two sides

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

KENDRIYA VIDYALAY SANGATHAN: CHENNAI REGION CLASS XII PRE-BOARD EXAMINATION Q.No. Value points Marks 1 0 ={0,2,4} 1.

KENDRIYA VIDYALAY SANGATHAN: CHENNAI REGION CLASS XII PRE-BOARD EXAMINATION Q.No. Value points Marks 1 0 ={0,2,4} 1. KENDRIYA VIDYALAY SANGATHAN: CHENNAI REGION CLASS XII PRE-BOARD EXAMINATION 7-8 Answer ke SET A Q.No. Vlue points Mrks ={,,4} 4 5 6.5 tn os.5 For orret proof 5 LHS M,RHS M 4 du dv os + / os. e d d 7 8

More information

PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by

PROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the

More information

Gauss Quadrature Rule of Integration

Gauss Quadrature Rule of Integration Guss Qudrture Rule o Integrtion Mjor: All Engineering Mjors Authors: Autr Kw, Chrlie Brker http://numerilmethods.eng.us.edu Trnsorming Numeril Methods Edution or STEM Undergrdutes /0/00 http://numerilmethods.eng.us.edu

More information

A Study on the Properties of Rational Triangles

A Study on the Properties of Rational Triangles Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn

More information

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

Area and Perimeter. Area and Perimeter. Solutions. Curriculum Ready.

Area and Perimeter. Area and Perimeter. Solutions. Curriculum Ready. Are n Perimeter Are n Perimeter Solutions Curriulum Rey www.mthletis.om How oes it work? Solutions Are n Perimeter Pge questions Are using unit squres Are = whole squres Are = 6 whole squres = units =

More information

HS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem

HS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem HS Pre-Alger Notes Unit 9: Roots, Rel Numers nd The Pythgoren Theorem Roots nd Cue Roots Syllus Ojetive 5.4: The student will find or pproximte squre roots of numers to 4. CCSS 8.EE.-: Evlute squre roots

More information

Mathematics 10 Page 1 of 5 Properties of Triangle s and Quadrilaterals. Isosceles Triangle. - 2 sides and 2 corresponding.

Mathematics 10 Page 1 of 5 Properties of Triangle s and Quadrilaterals. Isosceles Triangle. - 2 sides and 2 corresponding. Mthemtis 10 Pge 1 of 5 Properties of s Pthgoren Theorem 2 2 2 used to find the length of sides of right tringle Tpe of s nd Some s Theorems ngles s Slene Isoseles Equilterl ute - ll ngles re less thn 90

More information

] dx (3) = [15x] 2 0

] dx (3) = [15x] 2 0 Leture 6. Double Integrls nd Volume on etngle Welome to Cl IV!!!! These notes re designed to be redble nd desribe the w I will eplin the mteril in lss. Hopefull the re thorough, but it s good ide to hve

More information

Mathematics SKE: STRAND F. F1.1 Using Formulae. F1.2 Construct and Use Simple Formulae. F1.3 Revision of Negative Numbers

Mathematics SKE: STRAND F. F1.1 Using Formulae. F1.2 Construct and Use Simple Formulae. F1.3 Revision of Negative Numbers Mthemtis SKE: STRAND F UNIT F1 Formule: Tet STRAND F: Alger F1 Formule Tet Contents Setion F1.1 Using Formule F1. Construt nd Use Simple Formule F1.3 Revision of Negtive Numers F1.4 Sustitution into Formule

More information

Prerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) ,

Prerna Tower, Road No 2, Contractors Area, Bistupur, Jamshedpur , Tel (0657) , R Pen Towe Rod No Conttos Ae Bistupu Jmshedpu 8 Tel (67)89 www.penlsses.om IIT JEE themtis Ppe II PART III ATHEATICS SECTION I (Totl ks : ) (Single Coet Answe Type) This setion ontins 8 multiple hoie questions.

More information

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola. Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.

More information

MCH T 111 Handout Triangle Review Page 1 of 3

MCH T 111 Handout Triangle Review Page 1 of 3 Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:

More information

are coplanar. ˆ ˆ ˆ and iˆ

are coplanar. ˆ ˆ ˆ and iˆ SML QUSTION Clss XII Mthemtis Time llowed: hrs Mimum Mrks: Generl Instrutions: i ll questions re ompulsor ii The question pper onsists of 6 questions divided into three Setions, B nd C iii Question No

More information

Chapters Five Notes SN AA U1C5

Chapters Five Notes SN AA U1C5 Chpters Five Notes SN AA U1C5 Nme Period Section 5-: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles

More information

Something found at a salad bar

Something found at a salad bar Nme PP Something found t sld r 4.7 Notes RIGHT TRINGLE hs extly one right ngle. To solve right tringle, you n use things like SOH-H-TO nd the Pythgoren Theorem. n OLIQUE TRINGLE hs no right ngles. To solve

More information

Exam 1 Formula Sheet. du a2 u du = arcsin + C du a 2 + u 2 du = du u u 2 a 2 du =

Exam 1 Formula Sheet. du a2 u du = arcsin + C du a 2 + u 2 du = du u u 2 a 2 du = [rsin(u)] = u x [ros(u)] = u x u [rtn(u)] = + x [rot(u)] = u + x [rse(u)] = u x x [rs(u)] = u x x Exm Formul Sheet tn(u) u = ln os(u) + C ot(u) u = ln sin(u) + C se(u) u = ln se(u) + tn(u) + C s(u) u =

More information

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into

More information

A quick overview of the four conic sections in rectangular coordinates is presented below.

A quick overview of the four conic sections in rectangular coordinates is presented below. MAT 6H Rectngulr Equtions of Conics A quick overview of the four conic sections in rectngulr coordintes is presented elow.. Circles Skipped covered in previous lger course.. Prols Definition A prol is

More information

PROPERTIES OF TRIANGLES

PROPERTIES OF TRIANGLES PROPERTIES OF TRINGLES. RELTION RETWEEN SIDES ND NGLES OF TRINGLE:. tringle onsists of three sides nd three ngles lled elements of the tringle. In ny tringle,,, denotes the ngles of the tringle t the verties.

More information

Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.

Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4. Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix

More information

Introduction to Algebra - Part 2

Introduction to Algebra - Part 2 Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite

More information

QUADRATIC EQUATION. Contents

QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

21.1 Using Formulae Construct and Use Simple Formulae Revision of Negative Numbers Substitution into Formulae

21.1 Using Formulae Construct and Use Simple Formulae Revision of Negative Numbers Substitution into Formulae MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Contents STRAND G: Alger Unit 1 Formule Student Tet Contents Setion 1.1 Using Formule 1. Construt nd Use Simple Formule 1.3 Revision of Negtive Numers 1.4

More information

2.4 Linear Inequalities and Interval Notation

2.4 Linear Inequalities and Interval Notation .4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

More information

Lesson-5 PROPERTIES AND SOLUTIONS OF TRIANGLES

Lesson-5 PROPERTIES AND SOLUTIONS OF TRIANGLES Leon-5 PROPERTIES ND SOLUTIONS OF TRINGLES Reltion etween the ide nd trigonometri rtio of the ngle of tringle In ny tringle, the ide, oppoite to the ngle, i denoted y ; the ide nd, oppoite to the ngle

More information

AP Calculus AB Unit 4 Assessment

AP Calculus AB Unit 4 Assessment Clss: Dte: 0-04 AP Clulus AB Unit 4 Assessment Multiple Choie Identify the hoie tht best ompletes the sttement or nswers the question. A lultor my NOT be used on this prt of the exm. (6 minutes). The slope

More information

Math Lesson 4-5 The Law of Cosines

Math Lesson 4-5 The Law of Cosines Mth-1060 Lesson 4-5 The Lw of osines Solve using Lw of Sines. 1 17 11 5 15 13 SS SSS Every pir of loops will hve unknowns. Every pir of loops will hve unknowns. We need nother eqution. h Drop nd ltitude

More information