The Circle Higher Maths The Circle Information on

The Circle Higher Maths

The Circle Information on the items below General equation of a circle – completing the square General equation of a circle Introductory examples Circle examples 2 Exam level questions Tangents to circles 1 Tangents to circles 2 Circle examples 1 Circle examples 3 Basic Skills questions – exam level Problem solving questions 1 – exam level Problem solving questions 2 – exam level

Circle: - General equation ; completing the square. In each example, by completing the square on the x and the y terms and rearranging, show that the equations represent circles and state the centre and the radius of each. 1 2 3 4 5 6 7 8 9 10 11 12 x 2 + y 2 – 12 x – 4 y – 31 = 0 x 2 + y 2 + 4 x - 2 y + 4 = 0 x 2 + y 2 – 6 x – 8 y - 3 = 0 x 2 + y 2 – 2 x – 2 y – 2 = 0 x 2 + y 2 – 10 y - 24 = 0 x 2 + y 2 + 2 x - 2 y + 1 = 0 x 2 + y 2 + 20 x – 20 y + 119 = 0 x 2 + y 2 + 14 x – 8 y + 49 = 0 x 2 + y 2 – 16 x – 6 y + 59 = 0 x 2 + y 2 + 8 x - 8 y + 31 = 0 x 2 + y 2 – 4 x - 32 = 0 x 2 + y 2 - 6 x - 4 y + 14 = 0

General Equation of a Circle Check which of these equations represent circles. Where they do, find the centre and the radius of the circle. 1 2 3 4 5 6 7 8 9 10 11 12 x 2 + y 2 – 6 x – 4 y – 3 = 0 x 2 + y 2 – 2 x + 6 y + 6 = 0 x 2 + y 2 – 10 x – 8 y + 32 = 0 x 2 + y 2 – 4 x – 21 = 0 x 2 + y 2 + 2 x – 4 y + 1 = 0 x 2 + y 2 – 18 x + 2 y – 18 = 0 x 2 + y 2 + 12 x – 10 y + 12 = 0 x 2 + y 2 – 6 x – 6 y + 9 = 0 x 2 + y 2 – 121 = 0 x 2 + y 2 – 10 y + 21 = 0 x 2 + y 2 – 14 x + 6 y + 39 = 0 x 2 + y 2 + 2 x + 4 y + 7 = 0

Tangents to Circles - set 1 In each example check that the given point does lie on the circle and hence find the equation of the tangent to the circle at that point. Ex. Equation of circle Point 1 2 3 4 5 6 7 8 9 10 x 2 + y 2 = 25 x 2 + y 2 = 5 (x – 2)2 + (y – 3)2 = 10 x 2 + y 2 – 2 x – 4 y – 21 = 0 (x + 2)2 + y 2 = 26 x 2 + y 2 – 2 y – 1 = 0 x 2 + y 2 + 6 x – 2 y + 5 = 0 x 2 + (y + 2)2 = 4 x 2 + y 2 - 8 x + 2 y + 4 = 0 x 2 + y 2 - 2 x - 2 y - 23 = 0 (-3, 4) (1, 2) (5, 4) (6, 3) (3, 1) (1, 2) (-2, 3) (-2, -2) (6, 2) (5, -2)

Tangents to a circle at a point on the circle - set 2 In each example show that the point given lies on the circle and find the equation of the tangent to the circle at that point. Write the equation of the tangent in the form ax + by + c = 0 Question Equation of the circle Point 1 x 2 + y 2 = 5 (2, 1) 2 x 2 + y 2 = 13 (3, 2) 3 (x-2)2 + (y-1)2 = 13 (4, 4) 4 (x+3)2 + (y-2)2 = 2 (-2, 3) 5 (x-5)2 + (y-3)2 = 40 (7, 9) 6 (x-2)2 + y 2 = 45 (5, 6) 7 x 2 + y 2 - 2 x - 6 y – 7 = 0 (2, 7) 8 x 2 + y 2 + 2 x + 8 y – 41 = 0 (2, 3) 9 x 2 + y 2 – 10 x – 6 y + 29 = 0 (6, 5) 10 x 2 + y 2 – 4 x – 4 y + 3 = 0 (4, 3) 11 x 2 + y 2 – 12 y + 28 = 0 (2, 8) 12 x 2 + y 2 – 8 y + 3 = 0 (2, 7) 13 x 2 + y 2 – 4 x + 2 y – 32 = 0 (3, 5)

Circle Introductory examples 1. State the centre and radius of these circles : a) x 2 + y 2 = 49 b) 16 x 2 + 16 y 2 = 9 c) (x - 2)2 + (y - 3)2 = 1 d) x 2 + (y + 6)2 = 36 e) (x + 2)2 + (y - 1)2 = 25 f) (x + 1)2 + (y + 1)2 = 100 2. Write down the equations of circles with the following centres and radii : a) (2, 5) , r = 4 b) (-2, 7) , r = 5 c) (6, -2) , r = 3 3. a) Find the equation of the circle with centre the origin and passing through the point (-1, -6). b) Find the equation of the circle with centre (5, 4) and passing through (2, -1). 4. a) Find the equation of the circle passing through the points (0, 2), (2, 1) and (-1, 0). b) Find the equation of the circle passing through the points (0, 0), (-1, 1) and (2, 0). 5. If the point (h, 3) lies on the circle x 2 + y 2 + 4 x - y - 2 = 0 , find h.

6. Show that the point A(-2, 3) lies on the circle x 2 + y 2 + x - 4 y + 1 = 0 and find the equation of the tangent to the circle at A. 7. a) Find the points of intersection of the line y = x - 2 and the circle x 2 + y 2 + x + y - 6 = 0. b) Find the points of intersection of the line y = 3 x + 1 and the circle x 2 + y 2 + 4 x - 1 = 0. 8. Find the equation of the circle with centre C(2, -1) which passes through A(1, 1). Find the equation of the tangent at A and show that this tangent is also a tangent to the circle x 2 + y 2 -14 x + 2 y + 30 = 0. 9. Find the equation of the circle with centre P(5, -1) which passes through Q(-1, 2). Find the equation of the tangent at Q and show that this tangent passes through the centre of the circle x 2 + y 2 + 6 x + 4 y + 5 = 0. 10. Find the equation of the tangent to the circle x 2 + y 2 = 10 at the point A(1, -3). Show that this line is also a tangent to the circle x 2 + y 2 - 4 x - 8 y - 20 = 0 and find its point of contact.

11. i) Find the coordinates of the centre C and the radius of the circle whose equation is x 2 + y 2 - 2 x - 4 y - 3 = 0 ii) Show that the point A(3, 4) lies on the circle and find the equation of the tangent at A. iii) Show that the point P(7, 0) lies on this tangent. iv) Find the equation of the circle which passes through the points C, A and P. 12. A and B are the points (3/5, -21/5) and (3, -3) respectively. Show that A and B lie on the circle x 2 + y 2 - 4 x + 8 y + 18 = 0 and find the equations of the tangents at A and B. Solutions are on the next slide

Solutions 1. a) b) c) d) e) f) centre (0, 0) (2, 3) (0, -6) (-2, 1) (-1, -1) radius 7 3/4 1 6 5 10 2. a) (x - 2)2 + (y - 5)2 = 16 b) (x + 2)2 + (y - 7)2 = 25 c) (x - 6)2 + (y + 2)2 = 9 3. a) x 2 + y 2 = 37 b) (x - 5)2 + (y - 4)2 = 34 4. a) x 2 + y 2 - x - y - 2 = 0 b) x 2 + y 2 - 2 x - 4 y = 0 5. h = - 2 6. 3 x - 2 y + 12 = 0 7. a) (2, 0) (-1, -3) b) (0, 1) (-1, -2) 8. (x -2)2 + (y + 1)2 = 5 ; x - 2 y + 1 = 0 coincident point (5, 3) 9. (x - 5)2 + (y + 1)2 = 45 Tangent at Q has equation y = 2 x + 4. Centre (-3, -2) satisfies the equation. 10. Tangent at A has equation x = 3 y + 10. Coincident point (4, -2) 11. i) C(1, 2) r = 2 2 ii) tangent at A is y = - x + 7 iii) P(7, 0) satisfies the equation in ii) iv) (x - 4)2 + (y - 1)2 = 10 12. Tangent at A has equation y = -7 x Tangent at B has equation y = -x

Circle Examples (1) State the centre and the radius of each circle. [ Centre (-g, -f) Radius = ( g 2 + f 2 – c ) ] 1 2 3 4 5 6 7 8 9 10 11 12 x 2 + y 2 – 6 x – 4 y + 4 = 0 x 2 + y 2 – 2 x – 8 y + 16 = 0 x 2 + y 2 + 4 x + 4 y + 4 = 0 x 2 + y 2 - 12 x + 6 y + 35 = 0 x 2 + y 2 -20 x - 2 y + 99 = 0 x 2 + y 2 - 8 x = 0 x 2 + y 2 + 10 y + 17 = 0 x 2 + y 2 + 2 x - 2 y + 1 = 0 x 2 + y 2 - 14 x + 8 y + 56 = 0 x 2 + y 2 + 16 x + 61 = 0 x 2 + y 2 - 19 = 0 x 2 + y 2 - 24 x - 22 y + 263 = 0

Circle Examples (2) 1. In each example state the center and radius of the circle (x – 3)2 + ( y – 4 )2 = 25 (x + 4)2 + ( y – 2 )2 = 9 (x + 8)2 + ( y + 5 )2 = 144 (x – 5)2 + y 2 = 16 x 2 + ( y + 1 )2 = 49 (x + 3)2 + ( y – 7 )2 = 100 (x – 2. 5)2 + ( y – 4. 5 )2 = 2. 25 2. Write down the equations of the circles with the following centers and radius. Centre (2, 5) (3, -1) (-6, -2) (-4, 5) Radius 8 4 7 9

3. Do the following points lie in/on/outside the circle 4. with equation (x – 2)2 + ( y – 6 )2 = 34 a) (5, 8) b) (-2, 3) c) (7, 9) d) (3, 7) 4. Find the equation of the circle with center shown and passing through the point stated. Centre Passing through the point (2, 5) (3, 7) (-3, -4) (2, 6) (4, 1) (4, 4) 5. a) Find the equation of the tangent to the circle (x – 3) 2 + ( y – 4 )2 = 5 at the point (4, 6) on the circle. b) Find the equation of the tangent to the circle (x + 1) 2 + ( y – 3)2 = 8 at the point (1, 5) on the circle. c) Find the equation of the tangent to the circle (x – 2) 2 + ( y – 1 )2 = 40 at the point (-4, -1) on the circle.

d) Find the equation of the tangent to the circle e) x 2 + ( y – 2 )2 = 17 at the point (4, 3) on the circle. e) Find the equation of the tangent to the circle (x – 5) 2 + y 2 = 2 at the point (6, 1) on the circle.

Circle Examples (3) 1. Write down the equations of the circles with center the origin and 2. radius a) 12 b) 3 c) 20 d) 2. 5 2. Write down the radius of each of the circles a) x 2 + y 2 = 81 b) x 2 + y 2 = 144 c) x 2 + y 2 = 12 d) x 2 + y 2 = 0. 25 3. State whether the points below lie in/on/outside of the circle 4. with equation x 2 + y 2 = 60. a) (7, 4) b) (-3, 8) c) (1, 7) 4. Find the equations of the tangents to the circles at the point indicated. y y (4, 3) 0 0 x x (2, -5) 5. State the center and the radius of each of the circles a) x 2 + y 2 – 8 x – 10 y + 4 = 0 b) x 2 + y 2 –x –y - 1 = 0

y 6. Find the equation of the circle shown in the diagram. (4, 4) 0 x 7. Find the equation of the tangent to the circle (x - 1) 2 + ( y + 1 )2 = 8 at the point (3, 1) on the circle. 8. Explain why the equation x 2 + y 2 – 2 x – 4 y + 26 = 0 does 9. not represent a circle ? y 9. Find the equation of the circle shown in the diagram. (5, 0) x 10. Show that the line y = 3 x +10 is a tangent to the circle x 2 + y 2 – 8 x – 4 y – 20 = 0 and find the point of contact. 11. Show that the line y = 2 x – 5 is a tangent to the circle x 2 + y 2 = 5 and find the point of contact.

12. Find the equation of the tangent to the circle x 2 + y 2 = 10 at the point (-3, -1) on the circle. Show that this tangent is also a tangent to the circle x 2 + y 2 – 20 y + 60 = 0 and find the point of contact with this circle. 13. Show that the line x+9 = 0 is a tangent to the circles x 2 + y 2 = 81 and x 2 + y 2 + 10 x + 24 y + 153 = 0 and find the points of contact.

Circle – Exam level questions and Past paper questions. 1. Find the possible values of ‘k’ for which the 2. line y = x – k is a tangent to the circle x 2 + y 2 = 18. 2. An ear ring is to be made from silver wire and is designed in the shape of two circles with tangents to the outer circle as shown in the diagram on the left. The other diagram shows the ear ring related to the coordinate axis. The circles touch at (0, 0). The equation of the inner circle is x 2 + y 2 + 3 y = 0. The other circle intersects the y axis at (0, -4). The tangents meet the y axis at (0, -6). Find the total length of wire needed to make this ear ring. -4 -6

3. The diagram shows a ‘gingerbread man’ 14 cm high with a circular head and body. The equation of 14 the body is 2 + y 2 -10 x – 12 y + 45 = 0 x and the line of centres is parallel to the y axis. Find the equation of the head. 4. Show that the point (3, 1) lies on the circle with equation x 2 + y 2 – 4 x + 6 y – 4 = 0. Find the equation of the tangent to the circle at this point. 5. AB is a tangent at B to the circle shown. The circle centre C has equation (x-2)2 + (y-2)2 = 25. The point A has coordinates (10, 8). Find the area of triangle ABC y A B C O x

6. Show that the equation x 2 + y 2 + 2 x + 3 y + 5 = 0 does not represent a circle. 7. The straight line y = x cuts the circle x 2 + y 2 -6 x -2 y -24 = 0 at the points A and B. Find the coordinates of A and B Find the equation of the circle which has AB as diameter. y = x y A O B 8. Find the equation of the circle which has P(-2, -1) and Q(4, 5) as the end points of a diameter. x

9. The diagram shows three circles. The centres A, B and C are collinear i. e. they All lie on the same straight line. The equation of outer circles are (x+12)2+(y+15)2=25 and (x-24)2+(y-12)2=100. Find the equation of the central circle. y C O B A 10. Find the equation of the tangent to the circle x 2 + y 2 + 2 x – 4 y – 15 = 0 , at the point (3, 4) on the circle. 11. The line y = -1 is a tangent to the circle which passes through the points (0, 0) and (6, 0). Find the equation of this circle. x

12. Show that the line x + 3 y – 16 = 0 as a tangent to the circle x 2 + y 2 + 10 x + 6 y – 56 = 0. Find the point of contact. 13. Find the coordinates of the points where the line y = 2 x – 1 cuts 14. the circle with equation x 2 + y 2 = 2. B 14. In the diagram, AB and AC are the tangents from a point A(9, 0) to the circle with equation x 2 + y 2 = 16 with centre O. Find the area of the kite AOBC. A O C 15. Show that the line x + y = 10 is a tangent to the circle x 2 + y 2 – 2 x – 10 y + 18 = 0 and find the coordinates of the point of contact. 16. Find the centre and radius of the circle with equation x 2 + y 2 -6 x -8 y = 0. Find the equation of the tangent to this circle at the point (6, 8) on the circle.

17. Show that the equation of the circle which passes through the points (0, 0), (4, 0) and (0, -2) is x 2 + y 2 – 4 x + 2 y = 0. Show that the line y = 2 x – 10 is a tangent to this circle and find the point of contact. 18. Find the equation of the tangent to the circle x 2 + y 2 – 3 x + y – 16 = 0 at the point (4, 3). y 19. Two identical circles touch at the point P(9, 3) as shown in the diagram. One of the circles has equation x 2 + y 2 - 10 x – 4 y + 12 = 0. Find the equation of the other circle. P(9, 3) O 20. Find the coordinates of the points of intersection 21. of the circle with equation x 2 + y 2 + 10 x – 2 y - 14 = 0 and the line with equation y = 2 x + 1. x

21. The circle shown has equation (x-3)2 + (y+2)2 = 25 Find the equation of the tangent to this circle at the point (6, 2). Where does this tangent cut the y axis? y (6, 2) O x 22. The line with equation x – 3 y = k, is a tangent to the circle with equation x 2 + y 2 – 6 x + 8 y + 15 = 0. Find the possible values of ‘k’. 23. Show that the points (5, 4) , (-3, -2) and (5, -2) form a right angled triangle 24. and hence find the equation of the circle through these three points. 24. Show that the tangents to the circle x 2 + y 2 – 4 x - 2 y - 20 = 0 at 25. the points A(7, 1) and B(-1, 5) intersect at the point C(7, 11). 25. Find the equation of the circle which has centre (2, 3) and 26. which passes through the point (5, 6). a) Prove that the line x - y + 7 = 0 is a tangent to this circle. b) Show that where this line meets the circle x 2 + y 2 = r 2, x must satisfy the equation 2 x 2 + 14 x + (49 -r 2) = 0. c) Find the value of ‘r’ if the line x - y + 7 = 0 is to be a tangent d) to the circle x 2 + y 2 = r 2.

Solutions are on the next slide

Solutions to circle problem solving – sheet 1

Solutions are on the next slide

Solutions to circle problem solving – sheet 2

Information on each item is shown below. General equation of a circle – completing the square This contains examples which start with a general circle form and ends up with the (x-a)2+(y-b)2= r 2 General equation of a circle Starting from the general equation you have to work out the centre and radius Tangents to circles (set 1) Working out the equation of the tangent to a circle at a point on the circle Tangents to circles (set 2) More examples on working out the equation of the tangent to a circle at a point on the circle. Introductory examples A set of examples using all the forms of the equations of the circle. Continued on next slide

Circle Examples 1 Stating the centre and radius using the general equation. Circle Examples 2 Simple examples using the basic ideas of the circle. Circle Examples 3 More difficult examples using the basic ideas of the circle. Exam level questions A set of 25 exam level questions on the circle. Basic skills questions A set of basic skills questions on the circle. These questions are exam level. Problem solving questions – 2 sets Two sets of questions on the circle testing problem solving skills. These questions are exam level.