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
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 = 100 6 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
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) 8 2003 OL P2 Q3 2004 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. 9 2011 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
10 The centres and equations shown are: Centre Equation (0, 0) x 2 + y 2 = 4 (3, 2) x 3 2 + y 2 2 = 4 (3, 3) x 3 2 + y + 3 2 = 4 ( 2, 1) x + 2 2 + y + 1 2 = 4 ( 4, 2) x + 4 2 + 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 4 2 + y + 3 2 = 5 2 x 4 2 + y + 3 2 = 25 12 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
13 Write down the centre and radius of each of the following circles: a) x 3 2 + y 1 2 = 4 b) x 6 2 + y 2 2 = 16 c) x + 1 2 + y 5 2 = 49 d) x 7 2 + y + 2 2 = 100 e) x + 5 2 + y + 4 2 = 17 f) x 7 2 + y + 8 2 = 35 g) x 2 + y 3 2 = 14 14 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) 15 2004 OL P2 Q3 2005 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 + 4 2 + 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
16 2006 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). 17 2015 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
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 3 2 + 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
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
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
28 Substitute back into the line equation. Write down the resulting points. 29 2003 OL P2 Q3 2005 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. 30 2004 HL P2 Q3 [edited] The line L: x y 1 = 0 intersects the circle K: x 3 2 + 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
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 4 2 + 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 = 1 2 6 4 = 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 = 0 33 2006 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
34 2008 HL P2 Q1 2014 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 + 2 2 + 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
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 6 2 + 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 = 6 0 2 + 3 0 2 = 36 + 9 = 45 r C = 20, r D = 5 Internally: 20 5 = 5 45 Externally: 20 + 5 = 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 = 3 0 6 0 = 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 = 0 39 2012 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
40 2011 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