Recognise the Equation of a Circle. Solve Problems about Circles Centred at O. Co-Ordinate Geometry of the Circle - Outcomes
|
|
- Kory Rich
- 5 years ago
- Views:
Transcription
1 1 Co-Ordinate Geometry of the Circle - Outcomes Recognise the equation of a circle. Solve problems about circles centred at the origin. Solve problems about circles not centred at the origin. Determine whether a given point is inside or outside a circle. Solve problems about the intersection of lines and circles. Solve problems about tangents. Solve problems about intersecting circles. 2 Recognise the Equation of a Circle Recall that the equation of a line contains an x and a y. e.g. 2x + 4y = 7 is a line e.g. y = 3x 8 is a line Equations of circles have at least an x 2 and a y 2. They may also have an x and a y but these are not required. e.g. x 2 + y 2 = 6 is a circle e.g. x 2 + y 2 + 2x 6y + 7 = 0 is a circle e.g. x 2 + 7x + y 2 9 = 0 is a circle 3 Solve Problems about Circles Centred at O Circles are the locus of points which are one radius away from their centre. 1
2 4 What other formula has two squares adding to equal another square? Solve Problems about Circles Centred at O For a circle centred at the origin (0, 0), the equation is: Formula: x 2 + y 2 = r 2, where r is the radius of the circle. e.g. for radius 3: x 2 + y 2 = 3 2 x 2 + y 2 = 9 5 Solve Problems about Circles Centred at O Write down the equations of the circles centred at the origin with each of the following radiuses: a) radius = 3 b) radius = 4 c) radius = 5 d) radius = 6 e) radius = 8 f) radius = 10 g) radius = 25 Write down the radiuses from the following circle equations: h) x 2 + y 2 = 4 i) x 2 + y 2 = 16 j) x 2 + y 2 = 49 k) x 2 + y 2 = 5 l) x 2 + y 2 = 17 m) x 2 + y 2 = 8 n) x 2 + y 2 = Just like when you did this with lines! Solve Problems about Circles Centred at O The equation of a circle describes every point on the circumference. We can use the equation to find out if a point is on the circle or to figure out the radius. e.g. Which of the following circles is the point (3, 4) on? a) x 2 + y 2 = 0.75 b) x 2 + y 2 = 7 c) x 2 + y 2 = 12 d) x 2 + y 2 = 25 e) x 2 + y 2 = 34 2
3 7 x 2 + y 2 = r 2 Solve Problems about Circles Centred at O e.g. Find the radius and equation of the circles centred at the origin passing through the points: a) 4, 3 b) 5, 12 c) 1, 3 d) 4, 5 e) 3, 4 f) 24, 7 g) (3, 8) OL P2 Q OL P2 Q3 Solve Problems about Circles Centred at O a) The circle C has equation x 2 + y 2 = 25. i. Verify that ( 4, 3) is on the circle C. Write down the co-ordinates of a point that lies outside C and give a reason for your answer. a) The circle C has equation x 2 + y 2 = 36. i. Write down the radius of C. The radius of another circle is twice the radius of C. The centre of this circle is (0, 0). Write down its equation. b) A circle has equation x 2 + y 2 = 13. The points a(2, 3) and b 2, 3 are on the circle. Verify that [ab] is a diameter of the circle OL P2 Q2 Solve Problems about Circles Centred at O A circle c 1 has centre (0, 0) and diameter 8 units. a) Show c 1 on a co-ordinate diagram. b) Find the equation of c 1. c) Prove that the point (3, 2) is inside c 1 and that the point (3, 3) is outside it. 3
4 10 The centres and equations shown are: Centre Equation (0, 0) x 2 + y 2 = 4 (3, 2) x y 2 2 = 4 (3, 3) x y = 4 ( 2, 1) x y = 4 ( 4, 2) x y 2 2 = 4 What is the equation of a circle with centre (h, k) and radius r? 11 The equation of a circle with centre (h, k) and radius r is: Formula: x h 2 + y k 2 = r 2 This is one of the few situations where not distributing is considered simpler e.g. The circle c 1 has centre (4, 3) and radius 5. Its equation is therefore: x y = 5 2 x y = Write down the equations of the circles with the following centres and radiuses: Question Centre Radius a) (1, 5) 2 b) ( 3, 6) 4 c) (4, 2) 3 d) ( 1, 7) 7 e) (0, 3) 1 f) (0, 0) 5 4
5 13 Write down the centre and radius of each of the following circles: a) x y 1 2 = 4 b) x y 2 2 = 16 c) x y 5 2 = 49 d) x y = 100 e) x y = 17 f) x y = 35 g) x 2 + y 3 2 = Find the equations of the circles with given centres and passing through the given point: Question Centre Passing through a) (1, 5) (4, 9) b) ( 3, 6) (9, 1) c) (4, 2) (0, 0) d) ( 1, 7) ( 2, 2) e) (0, 3) (7, 4) f) (0, 0) ( 5, 2) OL P2 Q OL P2 Q3 K is a circle with centre ( 2, 1). It passes through the point ( 3, 4). i. Find the equation of K. The point t, 2t is on the circle K. Find the two possible values of t. The circle K has equation x y 3 2 = 36. i. Write down the co-ordinates of the centre of K. i The point (2, 3) is one end-point of a diameter of K. Find the co-ordinates of the other end-point. The point ( 4, y) is on the circle K. Find the two values of y. 5
6 OL P2 Q3 Recall corollary 3: each angle in a semicircle is a right angle The vertices of a right-angled triangle are p(1, 1), q(5, 1) and r(1, 4). The circle K passes through the points p, q and r. i. On a co-ordinate diagram, draw the triangle pqr. Mark the point c, the centre of K, and draw K. i Find the equation of K. Find the equation of K, the image of K under the translation 5, 1 (1, 4) OL P2 Q3 a) Draw the circle c: x 2 + y 2 = 25. Show your scale on both axes. b) Verify, using algebra, that A( 4, 3) is on c. c) Find the equation of the circle with centre ( 4, 3) that passes through the point (3, 4). 18 Determine if a Point is Inside or Outside c How do the points shown compare to the radius? 6
7 19 This works because the equation of a circle is essentially Pythagoras Theorem, which is also essentially the distance formula. Determine if a Point is Inside or Outside c The equation of a circle is x h 2 + y k 2 = r 2. A point on the circle will fit this equation. Any other point will be either too big or too small. For a point (x 1, y 1 ), the following applies: Equation x 1 h 2 + y 1 k 2 < r 2 x 1 h 2 + y 1 k 2 = r 2 x 1 h 2 + y 1 k 2 > r 2 Relationship to Circle Inside On Outside 20 Determine if a Point is Inside or Outside c Given a circle K: x 2 + y 2 = 36, determine whether the following points are inside, outside, or on K. a) 2, 4 b) ( 3, 3) c) 8, 3 d) 5, 0 e) 5, 3 f) 3.28, 2.34 g) 5, 3 h) (6, 4) 21 Determine if a Point is Inside or Outside c Given a circle c: x y 2 2 = 18, determine whether the following points are inside, outside, or on c. a) 6, 1 b) 0, 5 c) 4, 4 d) 2, 5 e) 1, 1 f) 1, 2 g) 11, 2 h) (3, 2) 7
8 22 Recall from co-ordinate geometry of the line, we can tell whether shapes intersect by: 1. Graphing them and seeing visually if / where they intersect, or 2. Solving their equations simultaneously and algebraically determining common solutions (i.e. intersections). e.g. Determine the intersection points of x 2 + y 2 = 25 and x y = 1 by drawing a graph. Verify your answer by finding the intersection points algebraically. 23 To graph the line and circle, we need points. For a circle, its centre and radius will allow us to draw it accurately. (centre (0, 0), radius 5) For a line (x y = 1), substitute values as before: x = 0 0 y = 1 y = 1 (0, 1) y = 0 x 0 = 1 x = 1 ( 1, 0) 24 8
9 25 To verify the answer algebraically, we require simultaneous equations. Elimination will not work for line-circle intersections. Substitution or transitivity are the only methods that work. x 2 + y 2 = 25 x y = 1 26 Rearrange the line equation to x = or y =. Substitute this into the circle equation. 27 Work through the algebra. Solve the quadratic at the end to get either x or y. 9
10 28 Substitute back into the line equation. Write down the resulting points OL P2 Q OL P2 Q3 The line x 2y + 5 = 0 intersects the circle x 2 + y 2 = 10 at the points a and b. i. Draw a co-ordinate diagram on graph paper showing the line, circle and the points of intersection. Verify your answer by finding the points of intersection algebraically. The line y = 10 2x intersects the circle x 2 + y 2 = 40 at the points a and b. i. Draw a co-ordinate diagram on graph paper showing the line, circle and the points of intersection. Verify your answer by finding the points of intersection algebraically HL P2 Q3 [edited] The line L: x y 1 = 0 intersects the circle K: x y 1 2 = 5. Find the co-ordinates of the points of intersection by drawing a graph and verify your answers by finding the intersection points algebraically. 10
11 31 Solve Problems about Tangents A tangent to a circle touches the circle at exactly one point. This is called the point of contact of the tangent. Theorem 20: Each tangent is perpendicular to the radius that goes to that point of contact. 32 Solve Problems about Tangents e.g. Find the equation of the tangent to the circle x y 2 2 = 13 at the point (6, 1). To get an equation of a line, we need a point and a slope. We have a point. The slope is perpendicular to the radius at the point of contact: centre: (4, 2); point of contact (6, 1) m R = y 2 y 1 x 2 x 1 = = 3 2 m T = m R = 2 3 y y 1 = m x x 1 y 1 = 2 3 x 6 3 y + 1 = 2 x 6 3y + 3 = 2x 12 2x 3y 15 = OL P2 Q3 Hint: if two tangents to the same circle are parallel, they must be at opposite ends of a diameter. Solve Problems about Tangents The circle C has equation x 2 + y 2 = 25 The line L is tangent to the circle C at the point ( 3, 4) i. Verify that the point ( 3, 4) is on C. i Find the slope of L. Find the equation of L. iv. The line T is another tangent to C and is parallel to L. Find the co-ordinates of the point at which T touches C. 11
12 HL P2 Q OL P2 Q3 Solve Problems about Tangents A tangent is drawn to the circle x 2 + y 2 = 13 at the point (2, 3). This tangent crosses the x-axis at (k, 0). Find the value of k. a) The circle c has equation x y 3 2 = 100. Write down the co-ordinates of A, the centre of c, and r, the radius of c. b) Show that the point P( 8, 11) is on the circle c. c) Find the slope of the radius [AP]. d) Hence, find the equation of t, the tangent to c at P. e) A second line k is tangent to c at the point Q and k t. Find the co-ordinates of Q. 35 Solve Problems about Intersecting Circles If two circles touch at a single point, their centres and the point of contact are collinear (on the same line). Circles may touch internally or externally and the distance between their centres, C 1 C 2 depends on their radiuses: 36 Solve Problems about Intersecting Circles Internally: C 1 C 2 = r 1 r 2 Externally: C 1 C 2 = r 1 + r 2 12
13 37 Just use a calculator for this Solve Problems about Intersecting Circles e.g. Two circles, C: x 2 + y 2 = 20 and D: x y 3 2 = 5 intersect at a single point P. a) Do the circles touch internally or externally? b) If P(4, 2), find the equation of their common tangent. C centre = (0, 0), D centre = (6, 3) CD = = = 45 r C = 20, r D = 5 Internally: 20 5 = 5 45 Externally: = 45 So they touch externally. 38 m = y 2 y 1 x 2 x 1 C centre 0, 0 D centre(6, 3) Solve Problems about Intersecting Circles b) Need a point and a slope. P(4, 2) is a point on the tangent. Slope is perpendicular to the radiuses. m CD = = 1 2 m T = 2 1 = 2 y y 1 = m x x 1 y 2 = 2 x 4 y 2 = 2x + 8 2x + y 10 = OL P2 Q4 Solve Problems about Intersecting Circles The diagram shows two circles c 1 and c 2 of equal radius. c 1 has centre (0, 0) and it cuts the x-axis at (5, 0). a) Find the equation of c 1. b) Show that the point P( 3, 4) is on c 1. c) The two circles touch at P( 3, 4). P is on the line joining the two centres. Find the equation of c 2. d) Find the equation of the common tangent at P. 13
14 OL P2 Q2 Solve Problems about Intersecting Circles A circle c 1 has centre (0, 0) and diameter 8 units. a) Show c 1 on a co-ordinate diagram. b) Find the equation of c 1. c) Prove that the point (3, 2) is inside c 1 and that the point 3, 3 is outside it. d) Another circle, c 2 has centre (0, 1) and just touches the circle c 1. Show c 2 on your diagram in part (a) and find the equation of c 2. 14
Edexcel New GCE A Level Maths workbook Circle.
Edexcel New GCE A Level Maths workbook Circle. Edited by: K V Kumaran kumarmaths.weebly.com 1 Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint
More informationDraft Version 1 Mark scheme Further Maths Core Pure (AS/Year 1) Unit Test 1: Complex numbers 1
1 w z k k States or implies that 4 i TBC Uses the definition of argument to write 4 k π tan 1 k 4 Makes an attempt to solve for k, for example 4 + k = k is seen. M1.a Finds k = 6 (4 marks) Pearson Education
More information+ 2gx + 2fy + c = 0 if S
CIRCLE DEFINITIONS A circle is the locus of a point which moves in such a way that its distance from a fixed point, called the centre, is always a constant. The distance r from the centre is called the
More informationMathematics. Single Correct Questions
Mathematics Single Correct Questions +4 1.00 1. If and then 2. The number of solutions of, in the interval is : 3. If then equals : 4. A plane bisects the line segment joining the points and at right angles.
More informationCircles, Mixed Exercise 6
Circles, Mixed Exercise 6 a QR is the diameter of the circle so the centre, C, is the midpoint of QR ( 5) 0 Midpoint = +, + = (, 6) C(, 6) b Radius = of diameter = of QR = of ( x x ) + ( y y ) = of ( 5
More information2. (i) Find the equation of the circle which passes through ( 7, 1) and has centre ( 4, 3).
Circle 1. (i) Find the equation of the circle with centre ( 7, 3) and of radius 10. (ii) Find the centre of the circle 2x 2 + 2y 2 + 6x + 8y 1 = 0 (iii) What is the radius of the circle 3x 2 + 3y 2 + 5x
More informationANALYTICAL GEOMETRY. Equations of circles. LESSON
7 LESSON ANALYTICAL GEOMETRY Analytical geometry in Gr12 mostly involves circles and tangents to circles. You will however need all the skills learnt in Gr11 to answer the questions. Equations of circles.
More informationCore Mathematics 2 Coordinate Geometry
Core Mathematics 2 Coordinate Geometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Coordinate Geometry 1 Coordinate geometry in the (x, y) plane Coordinate geometry of the circle
More informationChapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in
Chapter - 10 (Circle) Key Concept * Circle - circle is locus of such points which are at equidistant from a fixed point in a plane. * Concentric circle - Circle having same centre called concentric circle.
More informationMark scheme Pure Mathematics Year 1 (AS) Unit Test 2: Coordinate geometry in the (x, y) plane
Mark scheme Pure Mathematics Year 1 (AS) Unit Test : Coordinate in the (x, y) plane Q Scheme Marks AOs Pearson 1a Use of the gradient formula to begin attempt to find k. k 1 ( ) or 1 (k 4) ( k 1) (i.e.
More information1 k. cos tan? Higher Maths Non Calculator Practice Practice Paper A. 1. A sequence is defined by the recurrence relation u 2u 1, u 3.
Higher Maths Non Calculator Practice Practice Paper A. A sequence is defined b the recurrence relation u u, u. n n What is the value of u?. The line with equation k 9 is parallel to the line with gradient
More informationThis section will help you revise previous learning which is required in this topic.
Higher Portfolio Circle Higher 10. Circle Revision Section This section will help you revise previous learning which is required in this topic. R1 I can use the distance formula to find the distance between
More informationA-Level Notes CORE 1
A-Level Notes CORE 1 Basic algebra Glossary Coefficient For example, in the expression x³ 3x² x + 4, the coefficient of x³ is, the coefficient of x² is 3, and the coefficient of x is 1. (The final 4 is
More informationDISCRIMINANT EXAM QUESTIONS
DISCRIMINANT EXAM QUESTIONS Question 1 (**) Show by using the discriminant that the graph of the curve with equation y = x 4x + 10, does not cross the x axis. proof Question (**) Show that the quadratic
More informationA marks are for accuracy and are not given unless the relevant M mark has been given (M0 A1 is impossible!).
NOTES 1) In the marking scheme there are three types of marks: M marks are for method A marks are for accuracy and are not given unless the relevant M mark has been given (M0 is impossible!). B marks are
More informationQUESTION BANK ON STRAIGHT LINE AND CIRCLE
QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,
More informationUnit 3: Number, Algebra, Geometry 2
Unit 3: Number, Algebra, Geometry 2 Number Use standard form, expressed in standard notation and on a calculator display Calculate with standard form Convert between ordinary and standard form representations
More informationPart (1) Second : Trigonometry. Tan
Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,
More informationSecondary School Certificate Examination Syllabus MATHEMATICS. Class X examination in 2011 and onwards. SSC Part-II (Class X)
Secondary School Certificate Examination Syllabus MATHEMATICS Class X examination in 2011 and onwards SSC Part-II (Class X) 15. Algebraic Manipulation: 15.1.1 Find highest common factor (H.C.F) and least
More information(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2
CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5
More informationMEI Core 1. Basic Algebra. Section 1: Basic algebraic manipulation and solving simple equations. Manipulating algebraic expressions
MEI Core Basic Algebra Section : Basic algebraic manipulation and solving simple equations Notes and Examples These notes contain subsections on Manipulating algebraic expressions Collecting like terms
More informationAQA IGCSE FM "Full Coverage": Equations of Circles
AQA IGCSE FM "Full Coverage": Equations of Circles This worksheet is designed to cover one question of each type seen in past papers, for each AQA IGCSE Further Maths topic. This worksheet was automatically
More information9.7 Extension: Writing and Graphing the Equations
www.ck12.org Chapter 9. Circles 9.7 Extension: Writing and Graphing the Equations of Circles Learning Objectives Graph a circle. Find the equation of a circle in the coordinate plane. Find the radius and
More informationPLC Papers. Created For:
PLC Papers Created For: Algebra and proof 2 Grade 8 Objective: Use algebra to construct proofs Question 1 a) If n is a positive integer explain why the expression 2n + 1 is always an odd number. b) Use
More information1. Draw and label a diagram to illustrate the property of a tangent to a circle.
Master 8.17 Extra Practice 1 Lesson 8.1 Properties of Tangents to a Circle 1. Draw and label a diagram to illustrate the property of a tangent to a circle. 2. Point O is the centre of the circle. Points
More informationTrig Functions Learning Outcomes
1 Trig Functions Learning Outcomes Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem. Solve problems about trig functions in all quadrants of a unit
More information08/01/2017. Trig Functions Learning Outcomes. Use Trig Functions (RAT) Use Trig Functions (Right-Angled Triangles)
1 Trig Functions Learning Outcomes Solve problems about trig functions in right-angled triangles. Solve problems using Pythagoras theorem. Solve problems about trig functions in all quadrants of a unit
More informationCBSE Class X Mathematics Board Paper 2019 All India Set 3 Time: 3 hours Total Marks: 80
CBSE Class X Mathematics Board Paper 2019 All India Set 3 Time: 3 hours Total Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided
More informationLesson 18: Recognizing Equations of Circles
Student Outcomes Students complete the square in order to write the equation of a circle in center-radius form. Students recognize when a quadratic in xx and yy is the equation for a circle. Lesson Notes
More informationParabolas and lines
Parabolas and lines Study the diagram at the right. I have drawn the graph y = x. The vertical line x = 1 is drawn and a number of secants to the parabola are drawn, all centred at x=1. By this I mean
More informationTrans Web Educational Services Pvt. Ltd B 147,1st Floor, Sec-6, NOIDA, UP
Solved Examples Example 1: Find the equation of the circle circumscribing the triangle formed by the lines x + y = 6, 2x + y = 4, x + 2y = 5. Method 1. Consider the equation (x + y 6) (2x + y 4) + λ 1
More information( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x 2 = - 8y.
PROBLEMS 04 - PARABOLA Page 1 ( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x - 8. [ Ans: ( 0, - ), 8, ] ( ) If the line 3x 4 k 0 is
More informationP1 Chapter 6 :: Circles
P1 Chapter 6 :: Circles jfrost@tiffin.kingston.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 11 th August 2017 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More information5 Find an equation of the circle in which AB is a diameter in each case. a A (1, 2) B (3, 2) b A ( 7, 2) B (1, 8) c A (1, 1) B (4, 0)
C2 CRDINATE GEMETRY Worksheet A 1 Write down an equation of the circle with the given centre and radius in each case. a centre (0, 0) radius 5 b centre (1, 3) radius 2 c centre (4, 6) radius 1 1 d centre
More informationMATHEMATICS. Candidates exhibited the following as some of their strengths:
MATHEMATICS 1. STANDARD OF THE PAPER The standard of the paper compared favourably with that of previous years. Candidates performance this year was slightly better than that of previous years. 2. SUMMARY
More informationKing s Year 12 Medium Term Plan for LC1- A-Level Mathematics
King s Year 12 Medium Term Plan for LC1- A-Level Mathematics Modules Algebra, Geometry and Calculus. Materials Text book: Mathematics for A-Level Hodder Education. needed Calculator. Progress objectives
More informationNational Quali cations
H 08 X747/76/ National Quali cations Mathematics Paper (Non-Calculator) THURSDAY, MAY 9:00 AM 0:0 AM Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given only to
More informationSolved Paper SSC Maharashtra Exam March 207 Class - X Geometry Time : 2 Hours Max. Marks : 40 Note : (i) Solve all questions. Draw diagrams wherever necessary. (ii) Use of calculator is not allowed. (iii)
More informationDEPARTMENT OF MATHEMATICS
DEPARTMENT OF MATHEMATICS AS level Mathematics Core mathematics 1 C1 2015-2016 Name: Page C1 workbook contents Indices and Surds Simultaneous equations Quadratics Inequalities Graphs Arithmetic series
More informationCOURSE STRUCTURE CLASS -X
COURSE STRUCTURE CLASS -X Units Unit Name Marks I NUMBER SYSTEMS 06 II ALGEBRA 20 III COORDINATE GEOMETRY 06 IV GEOMETRY 15 V TRIGONOMETRY 12 VI MENSURATION 10 VII STATISTICS & PROBABILTY 11 Total 80 UNIT
More informationMATHEMATICS ( CANDIDATES WITH PRACTICALS/INTERNAL ASSESSMENT ) ( CANDIDATES WITHOUT PRACTICALS/INTERNAL ASSESSMENT )
Total No. of Printed Pages 6 X/5/M 0 5 MATHEMATICS ( CANDIDATES WITH PRACTICALS/INTERNAL ASSESSMENT ) Full Marks : 80 Pass Marks : 4 ( CANDIDATES WITHOUT PRACTICALS/INTERNAL ASSESSMENT ) Full Marks : 00
More informationCBSE X Mathematics 2012 Solution (SET 1) Section B
CBSE X Mathematics 01 Solution (SET 1) Section B Q11. Find the value(s) of k so that the quadratic equation x kx + k = 0 has equal roots. Given equation is x kx k 0 For the given equation to have equal
More informationPlane geometry Circles: Problems with some Solutions
The University of Western ustralia SHL F MTHMTIS & STTISTIS UW MY FR YUNG MTHMTIINS Plane geometry ircles: Problems with some Solutions 1. Prove that for any triangle, the perpendicular bisectors of the
More information4 The Trigonometric Functions
Mathematics Learning Centre, University of Sydney 8 The Trigonometric Functions The definitions in the previous section apply to between 0 and, since the angles in a right angle triangle can never be greater
More informationUNIT 3 CIRCLES AND VOLUME Lesson 1: Introducing Circles Instruction
Prerequisite Skills This lesson requires the use of the following skills: performing operations with fractions understanding slope, both algebraically and graphically understanding the relationship of
More informationAS PURE MATHS REVISION NOTES
AS PURE MATHS REVISION NOTES 1 SURDS A root such as 3 that cannot be written exactly as a fraction is IRRATIONAL An expression that involves irrational roots is in SURD FORM e.g. 2 3 3 + 2 and 3-2 are
More informationThe Not-Formula Book for C1
Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationName Period. Date: Topic: 9-2 Circles. Standard: G-GPE.1. Objective:
Name Period Date: Topic: 9-2 Circles Essential Question: If the coefficients of the x 2 and y 2 terms in the equation for a circle were different, how would that change the shape of the graph of the equation?
More informationCHAPTER ONE FUNCTIONS AND GRAPHS. In everyday life, many quantities depend on one or more changing variables eg:
CHAPTER ONE FUNCTIONS AND GRAPHS 1.0 Introduction to Functions In everyday life, many quantities depend on one or more changing variables eg: (a) plant growth depends on sunlight and rainfall (b) speed
More informationMath & 8.7 Circle Properties 8.6 #1 AND #2 TANGENTS AND CHORDS
Math 9 8.6 & 8.7 Circle Properties 8.6 #1 AND #2 TANGENTS AND CHORDS Property #1 Tangent Line A line that touches a circle only once is called a line. Tangent lines always meet the radius of a circle at
More informationMathematics AQA Advanced Subsidiary GCE Core 1 (MPC1) January 2010
Link to past paper on AQA website: http://store.aqa.org.uk/qual/gce/pdf/aqa-mpc1-w-qp-jan10.pdf These solutions are for your personal use only. DO NOT photocopy or pass on to third parties. If you are
More informationVerulam School Mathematics. Year 9 Revision Material (with answers) Page 1
Verulam School Mathematics Year 9 Revision Material (with answers) Page 1 Q1. (a) Simplify a 2 a 4 Answer... (b) Simplify b 9 b 3 Answer... (c) Simplify c 5 c c 5 Answer... (Total 3 marks) Q2. (a) Expand
More informationMATHS (O) NOTES. SUBJECT: Maths LEVEL: Ordinary Level TEACHER: Jean Kelly. The Institute of Education Topics Covered: Complex Numbers
MATHS (O) NOTES The Institute of Education 07 SUBJECT: Maths LEVEL: Ordinary Level TEACHER: Jean Kelly Topics Covered: COMPLEX NUMBERS Strand 3(Unit ) Syllabus - Understanding the origin and need for complex
More informationYEAR 10 PROGRAM TERM 1 TERM 2 TERM 3 TERM 4
YEAR 10 PROGRAM TERM 1 1. Revision of number operations 3 + T wk 2 2. Expansion 3 + T wk 4 3. Factorisation 7 + T wk 6 4. Algebraic Fractions 4 + T wk 7 5. Formulae 5 + T wk 9 6. Linear Equations 10 +T
More informationInversion Geometry on its head
Inversion Geometry on its head Bibliography: Geometry revisited Coxeter & Greitzer MAA 1967 Introduction to Geometry Coxeter Wiley Classics 1961 Algebraic Projective Geometry Semple & Kneebone OUP 1952
More informationNational Quali cations
H 2018 X747/76/11 National Quali cations Mathematics Paper 1 (Non-Calculator) THURSDAY, 3 MAY 9:00 AM 10:10 AM Total marks 60 Attempt ALL questions. You may NOT use a calculator. Full credit will be given
More informationConsider the equation different values of x we shall find the values of y and the tabulate t the values in the following table
Consider the equation y = 2 x + 3 for different values of x we shall find the values of y and the tabulate t the values in the following table x 0 1 2 1 2 y 3 5 7 1 1 When the points are plotted on the
More informationSOLVED SUBJECTIVE EXAMPLES
Example 1 : SOLVED SUBJECTIVE EXAMPLES Find the locus of the points of intersection of the tangents to the circle x = r cos, y = r sin at points whose parametric angles differ by /3. All such points P
More informationGRADE - 10 MATH WORKPLAN
GRADE - 10 MATH WORKPLAN 2018-19 DATES TOPIC DETAILS 5 th to 9 th Real numbers 1. Learn Divisibility of Integers (Euclid s Division Algorithm - Technique to compute HCF of two positive integers.) 2. Multiplication
More informationSYSTEM OF CIRCLES OBJECTIVES (a) Touch each other internally (b) Touch each other externally
SYSTEM OF CIRCLES OBJECTIVES. A circle passes through (0, 0) and (, 0) and touches the circle x + y = 9, then the centre of circle is (a) (c) 3,, (b) (d) 3,, ±. The equation of the circle having its centre
More informationClass X Delhi Math Set-3 Section A
Class X Delhi Math Set-3 Section A 1. The angle of depression of a car, standing on the ground, from the top of a 75 m high tower, is 30. The distance of the car from the base of the tower (in m.) is:
More informationFurther Mathematics Summer work booklet
Further Mathematics Summer work booklet Further Mathematics tasks 1 Skills You Should Have Below is the list of the skills you should be confident with before starting the A-Level Further Maths course:
More informationA2 HW Imaginary Numbers
Name: A2 HW Imaginary Numbers Rewrite the following in terms of i and in simplest form: 1) 100 2) 289 3) 15 4) 4 81 5) 5 12 6) -8 72 Rewrite the following as a radical: 7) 12i 8) 20i Solve for x in simplest
More informationy mx 25m 25 4 circle. Then the perpendicular distance of tangent from the centre (0, 0) is the radius. Since tangent
Mathematics. The sides AB, BC and CA of ABC have, 4 and 5 interior points respectively on them as shown in the figure. The number of triangles that can be formed using these interior points is () 80 ()
More information1 y = Recitation Worksheet 1A. 1. Simplify the following: b. ( ) a. ( x ) Solve for y : 3. Plot these points in the xy plane:
Math 13 Recitation Worksheet 1A 1 Simplify the following: a ( ) 7 b ( ) 3 4 9 3 5 3 c 15 3 d 3 15 Solve for y : 8 y y 5= 6 3 3 Plot these points in the y plane: A ( 0,0 ) B ( 5,0 ) C ( 0, 4) D ( 3,5) 4
More informationDistance in the Plane
Distance in the Plane The absolute value function is defined as { x if x 0; and x = x if x < 0. If the number a is positive or zero, then a = a. If a is negative, then a is the number you d get by erasing
More informationKey competencies (student abilities)
Year 9 Mathematics Cambridge IGCSE Mathematics is accepted by universities and employers as proof of mathematical knowledge and understanding. Successful Cambridge IGCSE Mathematics candidates gain lifelong
More informationCircles. Exercise 9.1
9 uestion. Exercise 9. How many tangents can a circle have? Solution For every point of a circle, we can draw a tangent. Therefore, infinite tangents can be drawn. uestion. Fill in the blanks. (i) tangent
More informationSURA's Guides for 3rd to 12th Std for all Subjects in TM & EM Available. MARCH Public Exam Question Paper with Answers MATHEMATICS
SURA's Guides for rd to 1th Std for all Subjects in TM & EM Available 10 th STD. MARCH - 017 Public Exam Question Paper with Answers MATHEMATICS [Time Allowed : ½ Hrs.] [Maximum Marks : 100] SECTION -
More informationConic Sections Session 2: Ellipse
Conic Sections Session 2: Ellipse Toh Pee Choon NIE Oct 2017 Toh Pee Choon (NIE) Session 2: Ellipse Oct 2017 1 / 24 Introduction Problem 2.1 Let A, F 1 and F 2 be three points that form a triangle F 2
More informationSystems of Nonlinear Equations and Inequalities: Two Variables
Systems of Nonlinear Equations and Inequalities: Two Variables By: OpenStaxCollege Halley s Comet ([link]) orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse.
More informationMATHEMATICS ( CANDIDATES WITH PRACTICALS/INTERNAL ASSESSMENT ) ( CANDIDATES WITHOUT PRACTICALS/INTERNAL ASSESSMENT )
Total No. of Printed Pages 6 X/3/M 0 3 MATHEMATICS ( CANDIDATES WITH PRACTICALS/INTERNAL ASSESSMENT ) Full Marks : 80 Pass Marks : 4 ( CANDIDATES WITHOUT PRACTICALS/INTERNAL ASSESSMENT ) Full Marks : 00
More informationQUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)
QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola) Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents
More information10.1 Tangents to Circles. Geometry Mrs. Spitz Spring 2005
10.1 Tangents to Circles Geometry Mrs. Spitz Spring 2005 Objectives/Assignment Identify segments and lines related to circles. Use properties of a tangent to a circle. Assignment: Chapter 10 Definitions
More informationMaths Assessment Framework Year 10 Higher
Success Criteria for all assessments: Higher Tier 90% 9 80% 8 70% 7 60% 6 50% 5 Please note the GCSE Mathematics is one of the first GCSEs which will be graded by number rather than A*, A, B, C etc. Roughly,
More informationCircle. Paper 1 Section A. Each correct answer in this section is worth two marks. 5. A circle has equation. 4. The point P( 2, 4) lies on the circle
PSf Circle Paper 1 Section A Each correct answer in this section is worth two marks. 1. A circle has equation ( 3) 2 + ( + 4) 2 = 20. Find the gradient of the tangent to the circle at the point (1, 0).
More informationUNIT 6. BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle. The Circle
UNIT 6 BELL WORK: Draw 3 different sized circles, 1 must be at LEAST 15cm across! Cut out each circle The Circle 1 Questions How are perimeter and area related? How are the areas of polygons and circles
More informationFrom the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot. Harish Chandra Rajpoot Rajpoot, HCR. Winter February 24, 2015
From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Winter February 24, 2015 Mathematical Analysis of Elliptical Path in the Annular Region Between Two Circles, Smaller Inside the Bigger One
More informationMAT100 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS
MAT100 OVERVIEW OF CONTENTS AND SAMPLE PROBLEMS MAT100 is a fast-paced and thorough tour of precalculus mathematics, where the choice of topics is primarily motivated by the conceptual and technical knowledge
More informationCALCULUS BASIC SUMMER REVIEW
NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=
More informationCBSE CLASS X MATH
CBSE CLASS X MATH - 2011 Q.1) Which of the following cannot be the probability of an event? A. 1.5 B. 3 5 C. 25% D. 0.3 Q.2 The mid-point of segment AB is the point P (0, 4). If the Coordinates of B are
More informationIndicate whether the statement is true or false.
PRACTICE EXAM IV Sections 6.1, 6.2, 8.1 8.4 Indicate whether the statement is true or false. 1. For a circle, the constant ratio of the circumference C to length of diameter d is represented by the number.
More information10. Show that the conclusion of the. 11. Prove the above Theorem. [Th 6.4.7, p 148] 4. Prove the above Theorem. [Th 6.5.3, p152]
foot of the altitude of ABM from M and let A M 1 B. Prove that then MA > MB if and only if M 1 A > M 1 B. 8. If M is the midpoint of BC then AM is called a median of ABC. Consider ABC such that AB < AC.
More informationSociety of Actuaries Leaving Cert Maths Revision 1 Solutions 19 November 2018
1. (Question 1, Paper 1, 2000) (a) 3x-5 + 1 = 3x 5 1 = 3x 6 = 3 (x-2) = 3 x-2 2-x = x-2 x-2 (x-2) (b) (c) Standard Factor Theorem Proof Let k be the third root so (x-t)²(x-k) = x³+ 3px + c (x²- 2tx + t²)(x-k)
More informationS56 (5.3) Recurrence Relations.notebook September 09, 2015
Daily Practice 31.8.2015 Q1. Write down the equation of a circle with centre (-1, 4) and radius 5 Q2. Given the circle with equation (x 4) 2 + (y + 5) 2 = 40. Find the equation of the tangent to this circle
More informationAEA 2007 Extended Solutions
AEA 7 Extended Solutions These extended solutions for Advanced Extension Awards in Mathematics are intended to supplement the original mark schemes, which are available on the Edexcel website.. (a The
More informationTime : 2 Hours (Pages 3) Max. Marks : 40. Q.1. Solve the following : (Any 5) 5 In PQR, m Q = 90º, m P = 30º, m R = 60º. If PR = 8 cm, find QR.
Q.P. SET CODE Q.1. Solve the following : (ny 5) 5 (i) (ii) In PQR, m Q 90º, m P 0º, m R 60º. If PR 8 cm, find QR. O is the centre of the circle. If m C 80º, the find m (arc C) and m (arc C). Seat No. 01
More informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More information2001 Higher Maths Non-Calculator PAPER 1 ( Non-Calc. )
001 PAPER 1 ( Non-Calc. ) 1 1) Find the equation of the straight line which is parallel to the line with equation x + 3y = 5 and which passes through the point (, 1). Parallel lines have the same gradient.
More informationUNIT I : NUMBER SYSTEMS
CLASS X First Term Marks : 80 UNITS MARKS I. NUMBER SYSTEMS 10 II. ALGEBRA 20 III. GEOMETRY 15 IV TRIGONOMETRY 20 V STATISTICS 15 TOTAL 80 UNIT I : NUMBER SYSTEMS 1. REAL NUMBERS (15) Periods Euclid's
More information3.2 Constructible Numbers
102 CHAPTER 3. SYMMETRIES 3.2 Constructible Numbers Armed with a straightedge, a compass and two points 0 and 1 marked on an otherwise blank number-plane, the game is to see which complex numbers you can
More informationMAT1035 Analytic Geometry
MAT1035 Analytic Geometry Lecture Notes R.A. Sabri Kaan Gürbüzer Dokuz Eylül University 2016 2 Contents 1 Review of Trigonometry 5 2 Polar Coordinates 7 3 Vectors in R n 9 3.1 Located Vectors..............................................
More informationC=2πr C=πd. Chapter 10 Circles Circles and Circumference. Circumference: the distance around the circle
10.1 Circles and Circumference Chapter 10 Circles Circle the locus or set of all points in a plane that are A equidistant from a given point, called the center When naming a circle you always name it by
More informationTest Corrections for Unit 1 Test
MUST READ DIRECTIONS: Read the directions located on www.koltymath.weebly.com to understand how to properly do test corrections. Ask for clarification from your teacher if there are parts that you are
More informationSolve problems involving tangents to a circle. Solve problems involving chords of a circle
8UNIT ircle Geometry What You ll Learn How to Solve problems involving tangents to a circle Solve problems involving chords of a circle Solve problems involving the measures of angles in a circle Why Is
More informationKENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32
KENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32 SAMPLE PAPER 05 (2018-19) SUBJECT: MATHEMATICS(041) BLUE PRINT : CLASS X Unit Chapter VSA (1 mark) SA I (2 marks) SA II (3 marks) LA (4 marks) Total Unit
More informationX- MATHS IMPORTANT FORMULAS SELF EVALUVATION
X- MATHS IMPORTANT FORMULAS SELF EVALUVATION 1. SETS AND FUNCTIONS 1. Commutative property i ii 2. Associative property i ii 3. Distributive property i ii 4. De Morgan s laws i ii i ii 5. Cardinality of
More informationCO-ORDINATE GEOMETRY. 1. Find the points on the y axis whose distances from the points (6, 7) and (4,-3) are in the. ratio 1:2.
UNIT- CO-ORDINATE GEOMETRY Mathematics is the tool specially suited for dealing with abstract concepts of any ind and there is no limit to its power in this field.. Find the points on the y axis whose
More informationCircles MODULE - II Coordinate Geometry CIRCLES. Notice the path in which the tip of the hand of a watch moves. (see Fig. 11.1)
CIRCLES Notice the path in which the tip of the hand of a watch moves. (see Fig..) 0 9 3 8 4 7 6 5 Fig.. Fig.. Again, notice the curve traced out when a nail is fied at a point and a thread of certain
More informationR1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member
Chapter R Review of basic concepts * R1: Sets A set is a collection of objects sets are written using set brackets each object in onset is called an element or member Ex: Write the set of counting numbers
More information