The Circle and The Line Richard Walsh 1 st December 2011
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1 The Circle and The Line Richard Walsh 1 st December 2011
2 Announcements If anyone has ed Kieran for last week s notes, they will be sent out on either Monday or Tuesday of next week. If there are any new people here, on the sign in sheet is the room number (eg. C1-061) you must attend for the workshop. Please note this room number as you MUST attend this room.
3 New Syllabus
4 Previous Question Concepts The Line Division of a line segment in a given ratio - Linear transformations -Area of a triangle -Theorem: Proof that f maps every pair of parallel lines to a pair of parallel lines. - Perpendicular lines -Calculating points where a line crosses x and y-axis -Area of a triangle - Linear transformations: Find the image of a line -Angle between 2 lines
5 Previous Question Concepts The Line Point of intersection of two lines -Equation of a line -Theorem: Proof of measure of angle between two lines with slopes m 1 and m 2 - Extension questions using this theorem -Linear transformations: Prove is a line - Parametric & Cartesian equations of a line - Distance between points -Linear transformations: finding the image of different points - Theorem: Perpendicular distance between a point and a line
6 Previous Question Concepts The Line Area of a triangle - Linear transformations: Images and ratios -Point of intersection of two lines -Slope - Perpendicular lines -Division of a line segment -Theorem: Proof of measure of angle between two lines with slopes m 1 and m 2 - Theorem application question
7 Summary of Concepts Checklist The Line Concept Divisionof a line in a given ratio Area of a triangle Angle between 2 lines Concurrencies of a triangle Perpendicular distance from a point to a line Parametric equations of a line Linear transformations Theorems or x
8 Division of a line in a given ratio P, and Q, can be divided internally or externally. Internal division: If R divides [PQ] internally in the ratio m:n Pg18 of tables,
9 The Line (a) P and Q are the points 1,4 and 3,7 respectively. Find the co-ordinates of the point that divides internally in the ratio 3:1. 1; 3; 4; 7 m=3; n=1 2,,,
10 2011 Q3(a)
11 2011 Q3(a)
12 Angle between 2 lines If two lines p and q have slopes and respectively, and is the angle between them, then: tan 1 Pg19 of tables
13 The Line 2009 (b) (i) Prove the formula (ii) Find the equations of the two lines that pass through the point (6,1) and make an angle of 45 with the line 2 0. (ii) Line: tan
14 2009 (b) Equation of : (6,1); : Equation of : (6,1); : 3 3 0
15 Parametric Equations Parametric equations are simply a representation in a different variable or number of variables (in leaving cert you only have to deal with one extra variable, usually t). You know that a line is given in the form 0. All that a parametric equation does is give us an x equation which defines the x co-ordinates you are plotting, and a y equation which defines the y co-ordinates.
16 Parametric Equations 2008 (a) The parametric equations 7 4and 3 3 represents a line, where. Find the Cartesian equation of the line
17 What is a linear transformation? People may be aware HOW to solve a linear transformation however you MAY not be certain about what they actually are. Just like central symmetry, axial symmetry and translations, linear combinations move points from one place to another. However, linear combinations use both the x and y value of the point being moved and sub them into a function involving x and y. For instance a line is mapped to a line. Distance, area and perpendicularityare three things which may not be conserved under linear transformations. (i.e. if perpendicularityis the case before the transformation, it may not exist after the transformation)
18 The Line 2010 (a) The line is perpendicular to the line Find the value of. (b) (i) The line 4 5 0cuts the x-axis at P and the y-axis at Q. Write down the co-ordinates of P and Q in terms of k. (ii) The area of the triangle OPQ is 10 square units, where O is the origin. Find the two possible values of k. (c) is the transformation,,, where and. The line has equation. (i) Find the equation of, the image of under. (ii) Find the value(s) of for which makes an angle of 45 with.
19 Previous Question Concepts The Circle Parametric and Cartesian Equations -Equation of a circle passing through 3 points - Intersection of circles - Tangents at points of intersection Equation of a circle with centre on point on circle - Find centre and radius when given equation -Tangent to a circle
20 Previous Question Concepts The Circle Show a point lies on a circle when given equation -Equation of a tangent -Equations of 2 circles when given points of intersection and distance to a common chord. Other years require similar knowledge and understanding.
21 Summary of Concepts Checklist The Circle Concept or x Equation of a circle (in various forms) (2011) Parametric equations of a circle (2011) Proving that a locus is a circle Intersection of a line and circle Finding the equation of a circle (geometric/algebraic approaches) Proving a line is a tangent to a circle Tangents parallel, or perpendicular, to a given line Equations oftangents from a point outside the circle Touching/Intersecting circles (2011) Chords
22 Tips Sketch whatever information the question is giving you. Visualising the problem is putting you far closer to solving it than if you just learn off steps! Think about how you might solve the problem once you have set it up visually (using formulae etc.) Try it out. When attempting these questions in class or in your study time ALWAYS try to visualise, sketch, and problem solve (DON T try to learn off steps!!!)
23 The Circle 2011 (a) The following parametric equations define a circle: 2 3sin, 3cos where What is the Cartesian equation of the circle? (b) Find the equation of the circle that passes through the points 0,3, 2,1 and 6,5. (c) The circle : has centre and radius. The circle : has centre B and radius. (i) Show that and intersect at two points. (ii) Show that the tangents to at these points of intersection pass through the centre of.
24 The Circle 2011 (a) Parametric equations again. This time trigonometric functions are given. We know we need the Cartesian form so we must find a way of manipulating the trigonometric parts in order to get just x s, y s and numbers. 2 3sin 2 3sin 3cos
25 The Circle 2011 (b)find the equation of the circle that passes through the points and
26 The Circle 2011 (b) We know that each of the three points given are on the circle. Therefore when we sub them into the general equation of a circle we will get zero as our answer. General Equation: , , ,
27 We must now solve for g,fand c
28 Equation of the circle is therefore: :
29 The Circle 2011 (c) The circle has centre and radius. The circle has centre B and radius. (i) Show that and intersect at two points.
30 2011 Q1(c) (i) The common chord can be found by subtracting the two equations of the circles
31 2011 Q1 (c) (i) Common chord: This is the equation of the common chord
32 2011 Q1 (c) (i) So now we have: , 7 2, ,1
33 2011 Q1 (c) (ii) Show that the tangents to at these points of intersection pass through the centre of.
34 2011 Q1(c)(ii) Equation of (green tangent): Find the slope between point of contact (-2,1) and the centre of (4,-1). The slope of is perpendicular to this slope 3 Equation of :3 0 We know that 2, , 7 :3 7 0
35 2011 Q1 (c)(ii) Going back to the question is the centre of on this line? Centre of 3, passes through the centre of. Now we must try this for the other tangent.
36 2011 Q1(c)(ii) Equation of (blue tangent): Find the slope between point of contact, of (4,-1). and the centre The slope of is perpendicular to this slope Equation of : We know that, , 57 :
37 2011 Q1(c)(ii) Finally is the centre of 3, 2 on this line? passes through the centre of. So the tangents to at the points of intersection pass through the centre of.
38 The Circle 2010 (a) A circle with centre 3, 4 passes through the point 7, 3. Find the equation of the circle. (b)(i) Find the centre and radius of the circle (ii) The line 3 4 0is a tangent to the circle Find the two possible values of. (c) A circle has the line 2 as a tangent at the point 2,4. The circle also passes through the point 4, 2. Find the equation of the circle.
39 2010 (a)
40 2010(b)(i)
41 2010 (b)(i)
42 2010 (b)(ii)
43 2010 (c)
44 2010 (c)
1. Use the Trapezium Rule with five ordinates to find an approximate value for the integral
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