Core 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 using the equation of a circle in the form (x a) 2 + (y b) 2 = r 2 and including use of the following circle properties: (i) the angle in a semicircle is a right angle; (ii) the perpendicular from the centre to a chord bisects the chord; (iii) the perpendicularity of radius and tangent. Students should be able to find the radius and the coordinates of the centre of the Circle given the equation of the circle, and vice versa. Core Mathematics 2 Coordinate Geometry 2

Finding the Midpoint of a Line To work out the midpoint of line we need to find the halfway point Midpoint of AB = (2,2) B (3,3) 2 The formula for the midpoint is:- A (1,1) 2 x + x 1 2, y + y 1 2 2 2 Where (x 1, y 1 ) and (x 2, y 2 ) are 2 given points on the line Example 1. If A(3,7) and B(11, 3) Find the midpoint of AB Midpoint of AB = x + x 1 2, y + y 1 2 2 2 = 3 + 11, 7 + 3 2 2 = 14 2, 4 2 = (7,2) Diameter of circles are often used in this topic because the midpoint will always be the centre the circle. Core Mathematics 2 Coordinate Geometry 3

Example 2. If A(2,3) and B is(5,9) and the centre of the circle. If AC is the diameter of the circle find the coordinates of C Midpoint of AB = x + x 1 2, y + y 1 2 2 2 (5, 9) = 2 + x, 3 + y 2 2 5 = 2 + x 2 and 9 = 3 + y 2 10 = 2 + x 18 = 3 + y 8 = x 15 = y C = (5,15) if x = 10 4 then y = x 1 y = 10 4 1 y = 3 5 The Centre of the circle is (2.5, 3.5) Core Mathematics 2 Coordinate Geometry 4

Chords and Perpendicular Lines A chord is a line that passes from one side of a circle to the other but which does not pass through the centre. A perpendicular line always cuts at 90. If it bisects a line then it cuts it exactly in half. It is often called a perpendicular bisector. When questions are talking about this then you need to use the equation of a normal and the midpoints. The perpendicular bisector of a chord always passes through the centre of a circle. The key to success is that you always need to draw a sketch so you know what is going on. Example 1. The Lines AB and CD are chords of a circle. The line y = 3x 11 is the perpendicular bisector of AB. The line y = x 1 is the perpendicular bisector of CD. Find the coordinates of the centre of the circle. We know the perpendicular bisector of a chord passes through the centre so the centre of the circle is where the lines meet! So solve simultaneously Core Mathematics 2 Coordinate Geometry 5

y = 3x 11 y = x 1 3x 11 = x 1 4x 11 = 1 4x = 10 x = 10 4 if x = 10 4 then y = x 1 y = 10 4 1 y = 3 5 The Centre of the circle is (2.5, 3.5) Core Mathematics 2 Coordinate Geometry 6

Distance Between Two Points To work out the distance between two points we use Pythagoras Midpoint of AB = (2,2) B (6,8) 4 A (3,4) 3 The formula for the distance between two points is:- (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Where (x 1, y 1 ) and (x 2, y 2 ) are 2 given points on the line Example 1. PQ is the diameter of a circle where p( 1,3) and Q(6, 3). Find the radius of the circle First we need to remember that Radius = half the Diameter PQ = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 PQ = (6 1) 2 + ( 3 3) 2 PQ = (7 2 + ( 6) 2 ) PQ = 85 Radius = 85 2 Angles in a semicircle An angle in a semicircle is always 90 when one side of the triangle is the diameter and all 3 sides sit on the circumference of the circle Core Mathematics 2 Coordinate Geometry 7

Example 2. The points A(2,6), B(5,7) and C(8, 2) lie on a circle. Show that ABC is a right angled triangle and find the area of the triangle Length AB = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 AB = (5 2) 2 + (7 6) 2 AB = 3 2 + 1 2 AB = 10 Length BC = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 BC = (8 5) 2 + ( 2 7) 2 BC = 3 2 + ( 9) 2 BC = 90 Length AC = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 AC = (8 2) 2 + ( 2 6) 2 AC = 6 2 + ( 8) 2 AC = 100 AC = 10 Using pythagoras to prove ABC is a right angled triangle AC 2 = AB 2 + BC 2 10 2 = ( 10) 2 + ( 90) 2 100 = 10 + 90 100 = 100 This proves the triangle is a right angled triangle Core Mathematics 2 Coordinate Geometry 8

Area of a triangle = 1 2 base height A = 1 2 AB BC A = 1 2 10 90 A = 1 2 10 9 10 A = 15 units 2 Core Mathematics 2 Coordinate Geometry 9

Equation of a Circle An equation of a circle is always in the form (x a) 2 + (y b) 2 = r 2 where r is the radius and (a,b) is the centre of the circle. Example 1. If a circle has a radius of 7 and a centre at (2,6), what is the equation of the circle? (x a) 2 + (y b) 2 = r 2 where a = 2, b = 6, r = 7 (x 2) 2 + (y 6) 2 = 7 2 (x 2) 2 + (y 6) 2 = 49 The equation of the cirlce is (x 2) 2 + (y 6) 2 = 49 Example 2. Given the equation (x 2 3) 2 + (y + 7) 2 = 144, find the radius of the centre of the circle. (x a) 2 + (y b) 2 = r 2 where a = 2 3, b = 7, r 2 = 144 centre is (2 3, 7) r 2 = 144 r = 12 as radius cannot be negative we can ingnore the negative value r = 12 Example 3. equation Prove that (1,2) lies on the circumference of the circle which has the (x 2) 2 + (y + 3) 2 = 26 when x = 1 y = 2 (1 2) 2 + (2 + 3) 2 = 26 ( 1) 2 + (5) 2 = 26 1 + 25 = 26 26 = 26 (1,2) lies on the circumference of the circle Core Mathematics 2 Coordinate Geometry 10

The angle between the tangent and a radius is 90. A tangent only touches at one point. This circle theorem is often used in questions as it can relate closely to perpendicular bisectors. Tangents Example 1. The line 4x 3y 40 = 0 touches the circle (x 2) 2 + (y 6) 2 = 100 at P(10,0). Show that the radius at P is perpendicular to the line. This mean the centre A is (2,6) Gradient of AP = y 2 y 1 x 2 x 1 A = 6 1 2 10 = 6 8 P (10,0) gradient of AP is 3 4 Gradient of tangent 4x 3y 40 = 0 4x 40 = 3y 4x 40 3 = y gradient of tangent is 4 3 Using m 1 m 2 = 1 where m 1 = 3 4 and m 2 = 4 3 3 4 4 3 = 1 lines are perpendicualr Core Mathematics 2 Coordinate Geometry 11

Finding Points of Intersection If you need to find where a circle meets a line then solve the two equations simultaneously. Example 1. Find where the line y = x + 5 meets the circle x 2 + (y 2) 2 = 29 Substitute y = x + 5 into x 2 + (y 2) 2 = 29 x 2 + ((x + 5) 2) 2 = 29 x 2 + (x + 3) 2 = 29 x 2 + x 2 + 6x + 9 = 29 2x 2 + 6x 20 = 0 x 2 + 3x 10 = 0 (x + 5)(x 2) = 0 x = 5 or x = 2 if x = 5 y = x + 5 y = 5 + 5 y = 0 ( 5,0) if x = 2 y = x + 5 y = 2 + 5 y = 7 (2,7) So the line meets the circle at (-5,0) and (2,7). If you get no solutions when you try and solve two equations then it means the lines do not meet Core Mathematics 2 Coordinate Geometry 12

(C2-4.1) Name: Homework Questions 1 Midpoint of a Line 1. If A (9,15) and B (13,21). Find the midpoint of the line AB 2. If CD is the diameter of a circle, where C(2,6) and D (8,14) respectively. Find the coordinates of the centre of the circle. 3. If AB is the diameter of a circle and A(a, 4b) and B(3a, 3b+1) Find the value of a and b given that the centre of the circle is (4,11) 4. The line RS is a diameter of a circle centre (0,4). Given R is (3,10), find the coordinate of S 5. Look at the diagram below. Find the midpoint of AB B (2,15) C (2,7) A( 6,7) Core Mathematics 2 Coordinate Geometry 13

(C2-4.2) Name: Homework Questions 2 Perpendicular Bisectors from Chords 1. The Line PQ is a diameter of a circle centre C where P(1,2) and Q(3,4). The line T passes through C and is perpendicular to PQ. a) Find the midpoint of PQ b) Find the gradient of PQ c) Hence find the equation of T 2. The Line RS is a diameter of a circle centre P where R(-1,-2) and S(6,-5). The Line L passes through P and is perpendicular to RS, find the equations of L in the form ax+by+c=0, where a,b and c are integers. Core Mathematics 2 Coordinate Geometry 14

3. The Line AB is a chord on the circle with centre (3,-2). If A(6,-2) and B(3,1). The line L is perpendicular to AB and bisects it. Find the equation of L give your answer in the form y=mx+c 4. The points R(-2,5) S(2,1) T(-6,1) lie on the circumference of a circle. Find the equation of RS and RT and hence find the coordinate of the centre of the circle Core Mathematics 2 Coordinate Geometry 15

Core Mathematics 2 Coordinate Geometry 16

(C2-4.3) Name: Homework Questions 3 Distance Between 2 Points 1. If P(3,-2) and Q(-4,-3). Find the distance between PQ 2. If A (2,6) and B (8,14). Find the distance between AB 3. If CD is the diameter of a circle where C (3, -4), and D(2, 5). Find the radius of the circle. 4. The point (-4,9) lies on the circle centre (-4,3). Find the diameter of the circle 5. Points A(2,-1) and B(-2,-5) lie on the circumference of a circle centre C(2,-5). a) Prove if ABC is a right angled triangle or not b) Calculate the diameter of the circle Core Mathematics 2 Coordinate Geometry 17

12 34 5 12 34 5 12 34 5 12 34 5 C2-4.4) Name: Homework Questions 4 Equations of Circles 1. Plot the following graphs on the axis below a) x 2 + y 2 = 16 y 5 b) x 2 + y 2 = 4 4 3 2 1 5 4 3 2 1 1 1 2 3 4 5 x 2 3 4 5 2. Find the coordinate of the center of the following circle a) x 2 + y 2 = 36 b) (x 3) 2 + y 2 = 56 c) (x + 2) 2 + (y 5) 2 = 76 d) x 2 + (y + 7) 2 = 58 3. Find the length of the radius of the following circles a) x 2 + y 2 = 100 Core Mathematics 2 Coordinate Geometry 18

b) (x + 2) 2 + (y 3) 2 = 49 48 12 48 12 48 12 48 12 c) (x 5) 2 + (y + 8) 2 = 30 4. Draw the graph and state the centre and radius of the circle (x 3) 2 + (y 1) 2 = 36 y 12 8 4 12 8 4 4 8 12 x 4 24 68 24 68 24 68 24 68 8 12 5. Draw the graph of (x + 2) 2 + y 2 = 16 y 8 6 4 2 8 6 4 2 2 2 4 6 8 x 4 6 8 6. Show that the following circle passess through the point (5,9) (x 3) 2 + (y 4) 2 = 29 Core Mathematics 2 Coordinate Geometry 19

7. The point P(2,-3) lies on the circle centre (7,5). a) Find the equation of the circle b) Find the equation of the tangent to the circle at P Core Mathematics 2 Coordinate Geometry 20

C2-4.5) Name: Homework Questions 5 Finding Points of Intersections for Circles 1. Find where the circles below meet the x-axis a) (x + 2) 2 + (y 5) 2 = 50 b) ((x 3) 2 + (y + 6) 2 = 52 2. Find where the circles below meet the y-axis a) (x 3) 2 + (y 1) 2 = 10 b) (x + 4) 2 + (y 5) 2 = 25 3. Find the points of intersection of the following line and circle x 2 + y 2 = 4 y = x + 2 4. Show that the line below is a tangent to the circle y = 5 (x 1) 2 + (y 2) 2 = 9 Core Mathematics 2 Coordinate Geometry 21

Past examination questions 1. The points A and B have coordinates (5, 1) and (13, 11) respectively. (a) Find the coordinates of the mid-point of AB. Given that AB is a diameter of the circle C, (b) find an equation for C. (4) (C2, Jan2005 Q2) 2. The circle C, with centre at the point A, has equation x 2 + y 2 10x + 9 = 0. Find (a) the coordinates of A, (b) the radius of C, (c) the coordinates of the points at which C crosses the x-axis. Given that the line l with gradient 2 7 is a tangent to C, and that l touches C at the point T, (d) find an equation of the line which passes through A and T. (3) (C2, June2005 Q8) Core Mathematics 2 Coordinate Geometry 22

3. Figure 1 y B C P O A x In Figure 1, A(4, 0) and B(3, 5) are the end points of a diameter of the circle C. Find (a) the exact length of AB, (b) the coordinates of the midpoint P of AB, (c) an equation for the circle C. (3) (C2, Jan2006 Q3) 4. The line joining points ( 1, 4) and (3, 6) is a diameter of the circle C. Find an equation for C. (6) (C2, Jan2007 Q3) Core Mathematics 2 Coordinate Geometry 23

5. Figure 1 y y = 3x 4 P(2, 2) C Q O x The line y = 3x 4 is a tangent to the circle C, touching C at the point P(2, 2), as shown in Figure 1. The point Q is the centre of C. (a) Find an equation of the straight line through P and Q. (3) Given that Q lies on the line y = 1, (b) show that the x-coordinate of Q is 5, (c) find an equation for C. (1) (4) (C2, May 2006 Q7) 6. y B Core Mathematics 2 Coordinate Geometry 24 M (3, 1)

Figure 3 The points A and B lie on a circle with centre P, as shown in Figure 3. The point A has coordinates (1, 2) and the mid-point M of AB has coordinates (3, 1). The line l passes through the points M and P. (a) Find an equation for l. (4) Given that the x-coordinate of P is 6, (b) use your answer to part (a) to show that the y-coordinate of P is 1, (c) find an equation for the circle. (1) (4) (C2, May 2007 Q7) Core Mathematics 2 Coordinate Geometry 25

7. A circle C has centre M(6, 4) and radius 3. (a) Write down the equation of the circle in the form (x a) 2 + (y b) 2 = r 2. y T Figure 3 C 3 M (6, 4) Q P (12, 6) x Figure 3 shows the circle C. The point T lies on the circle and the tangent at T passes through the point P (12, 6). The line MP cuts the circle at Q. (b) Show that the angle TMQ is 1.0766 radians to 4 decimal places. (4) The shaded region TPQ is bounded by the straight lines TP, QP and the arc TQ, as shown in Figure 3. (c) Find the area of the shaded region TPQ. Give your answer to 3 decimal places. (5) (C2, Jan 2008 Q8) 8. The circle C has centre (3, 1) and passes through the point P(8, 3). (a) Find an equation for C. (b) Find an equation for the tangent to C at P, giving your answer in the form ax + by + c = 0, where a, b and c are integers. (5) Core Mathematics 2 Coordinate Geometry 26 (4) C2, June 2008 Q5)

9. Figure 2 The points P( 3, 2), Q(9, 10) and R(a, 4) lie on the circle C, as shown in Figure 2. Given that PR is a diameter of C, (a) show that a = 13, (b) find an equation for C. 10. The circle C has equation (3) (5) (C2, Jan 2009 Q5) x 2 + y 2 6x + 4y = 12 (a) Find the centre and the radius of C. (5) The point P( 1, 1) and the point Q(7, 5) both lie on C. (b) Show that PQ is a diameter of C. The point R lies on the positive y-axis and the angle PRQ = 90. (c) Find the coordinates of R. (4) (C2, June 2009 Q6) Core Mathematics 2 Coordinate Geometry 27

11. y C O N x A 12 B P Figure 3 Figure 3 shows a sketch of the circle C with centre N and equation (a) Write down the coordinates of N. (x 2) 2 + (y + 1) 2 = 169. 4 (b) Find the radius of C. (1) The chord AB of C is parallel to the x-axis, lies below the x-axis and is of length 12 units as shown in Figure 3. (c) Find the coordinates of A and the coordinates of B. (5) (d) Show that angle ANB = 134.8, to the nearest 0.1 of a degree. The tangents to C at the points A and B meet at the point P. (e) Find the length AP, giving your answer to 3 significant figures. (C2, Jan 2010 Q8) Core Mathematics 2 Coordinate Geometry 28

12. The circle C has centre A(2,1) and passes through the point B(10, 7). (a) Find an equation for C. (4) The line l 1 is the tangent to C at the point B. (b) Find an equation for l 1. (4) The line l 2 is parallel to l 1 and passes through the mid-point of AB. Given that l 2 intersects C at the points P and Q, (c) find the length of PQ, giving your answer in its simplest surd form. (3) (C2, June 2010 Q10) 13. The points A and B have coordinates ( 2, 11) and (8, 1) respectively. Given that AB is a diameter of the circle C, (a) show that the centre of C has coordinates (3, 6), (b) find an equation for C. (c) Verify that the point (10, 7) lies on C. (d) Find an equation of the tangent to C at the point (10, 7), giving your answer in the form y = mx + c, where m and c are constants. (4) (C2, Jan 2011 Q9) (1) (4) (1) 14. The circle C has equation x 2 + y 2 + 4x 2y 11 = 0. Find (a) the coordinates of the centre of C, (b) the radius of C, (c) the coordinates of the points where C crosses the y-axis, giving your answers as simplified surds. (4) (C2, May2011 Q4) Core Mathematics 2 Coordinate Geometry 29

15. A circle C has centre ( 1, 7) and passes through the point (0, 0). Find an equation for C. (4) (C2, Jan 2012 Q2) 16. Figure 1 The circle C with centre T and radius r has equation x 2 + y 2 20x 16y + 139 = 0. (a) Find the coordinates of the centre of C. (b) Show that r = 5 (3) The line L has equation x = 13 and crosses C at the points P and Q as shown in Figure 1. (c) Find the y coordinate of P and the y coordinate of Q. (3) Given that, to 3 decimal places, the angle PTQ is 1.855 radians, (d) find the perimeter of the sector PTQ. (3) (C2, May 2012 Q3) Core Mathematics 2 Coordinate Geometry 30

17. The circle C has equation x 2 + y 2 20x 24y + 195 = 0. The centre of C is at the point M. (a) Find (i) the coordinates of the point M, (ii) the radius of the circle C. (5) N is the point with coordinates (25, 32). (b) Find the length of the line MN. The tangent to C at a point P on the circle passes through point N. (c) Find the length of the line NP. 18. (C2, Jan 2013 Q5) Figure 4 The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4. (a) Write down an equation for the circle C, that is shown in Figure 4. (3) Core Mathematics 2 Coordinate Geometry 31

A line through the point P(8, 7) is a tangent to the circle C at the point T. (b) Find the length of PT. 19. The circle C, with centre A, passes through the point P with coordinates ( 9, 8) and the point Q with coordinates (15, 10). (3) (C2, May 2013 Q10) Given that PQ is a diameter of the circle C, (a) find the coordinates of A, (b) find an equation for C. A point R also lies on the circle C. Given that the length of the chord PR is 20 units, (3) 20. (c) find the length of the shortest distance from A to the chord PR. Give your answer as a surd in its simplest form. (d) Find the size of the angle ARQ, giving your answer to the nearest 0.1 of a degree. (C2, May 20134_R Q10) Figure 3 Figure 3 shows a circle C with centre Q and radius 4 and the point T which lies on C. The tangent to C at the point T passes through the origin O and OT = 6 5. Given that the coordinates of Q are (11, k), where k is a positive constant, Core Mathematics 2 Coordinate Geometry 32

(a) find the exact value of k, (b) find an equation for C. 21. A circle C with centre at the point (2, 1) passes through the point A at (4, 5). (a) Find an equation for the circle C. (3) (C2, May 2014 Q9) (3) (b) Find an equation of the tangent to the circle C at the point A, giving your answer in the form ax + by + c = 0, where a, b and c are integers. (4) (C2, May 2015 Q2) Core Mathematics 2 Coordinate Geometry 33