hsn.uk.net Circles Contents Circles 1 1 Representing a Circle A 1 Testing a Point A 3 The General Equation of a Circle A 4 Intersection of a Line and a Circle A 4 5 Tangents to Circles A 5 6 Equations of Tangents to Circles A 6 7 Intersection of Circles A 8 1
Circles 1 Representing a Circle A The equation of a circle with centre ( ab, ) and radius r units is: ( x a) + ( y b) = r. This is given in the exam. For example, the circle with centre (, 1) and radius 4 units has equation: ( x ) ( y ) ( x ) ( y ) + + 1 = 4 + + 1 = 16. Note that the equation of a circle with centre ( 0, 0 ) is of the form x + y = r, where r is the radius of the circle. EXAMPLES 1. Find the equation of the circle with centre ( 1, 3) and radius 3 units. ( ) ( ) x a + y b = r ( ) ( ) ( x 1) + y ( 3) = 3 ( x ) ( y ) 1 + + 3 = 3.. A is the point ( 3,1) and B( 5,3 ). Find the equation of the circle which has AB as a diameter. The centre of the circle is the midpoint of AB; 5 33+ 1 C = midpoint ( ) AB =, = 1,. The radius r is the distance between A and C: r = ( 1 ( 3) ) + ( 1) = 4 + 1 = 17. So the equation of the circle is ( ) ( ) x 1 + y = 17. Note You could also use the distance between B and C, or halve the distance between A and B. hsn.uk.net
Testing a Point A Given a circle with centre ( ab, ) and radius r units, we can determine whether a point ( pq, ) lies within, outwith or on the circumference using the following rules: ( ) ( ) ( ) ( ) ( ) ( ) p a + q b < r the point lies within the circle p a + q b = r the point lies on the circumference of the circle p a + q b > r the point lies outwith the circle. EXAMPLE A circle has the equation ( ) ( ) x + y + 5 = 9. Determine whether the points (,1 ), ( 7, 3) and ( 3, 4) lie within, outwith or on the circumference of the circle. Point (,1 ) : x + y + 3 ( ) ( ) = ( ) + ( 1+ 5) = 0 + 6 = 36 > 9 Point ( 7, 3) : ( x ) + ( y + 3) = ( 7 ) + ( 3+ 5) = 5 + = 9 Point ( 3, 4) : ( x ) + ( y + 3) ( 3 ) ( 4 5) = + + = 1 + 1 = < 9 So outwith the circle. So on the circumference. So within the circle. 3 The General Equation of a Circle A The equation of any circle can be written in the form where the centre is ( g, f ) This is given in the exam. x y gx fy c + + + + = 0 and the radius is g + f c units. Note that the above equation only represents a circle if g + f c > 0, since: if g + f c < 0 then we cannot obtain a real value for the radius, since we would have to square root a negative; if g + f c = 0 then the radius is zero the equation represents a point rather than a circle. hsn.uk.net 3
EXAMPLE 1. Find the radius and centre of the circle with equation x + y + 4x 8y + 7= 0. Comparing with g = 4 so g = f = 8 so f = 4 c = 7 x y gx fy c + + + + = 0: Centre is, ( g f ) = (, 4) Radius is. Find the radius and centre of the circle with equation x + y 6x + 10y = 0. g + f c ( ) 4 7 = + = 4 + 16 7 = 13 units. The equation must be in the form x + y + gx + fy + c = 0, so divide each term by : x y x y + 3 + 5 1= 0 Now compare with g = 3 so g = 3 f = 5 so f = 5 c = 1 3. Explain why x y gx fy c + + + + = 0: Centre is, x y x y ( g f ) 3 5 (, ) = Radius is g + f c 3 5 ( ) ( ) = + + 1 = = 9 + 5 + 4 4 4 4 38 4 = 38 units. + + 4 8 + 9 = 0 is not the equation of a circle. Comparing with x + y + gx + fy + c = 0: g = 4 so g = g + f c = + ( 4) 9 f = 8 so f = 4 = 9 < 0. c = 9 The equation does not represent a circle since g + f c > 0 is not satisfied. hsn.uk.net 4
4. For which values of k does a circle? Comparing with x y kx y k k x y gx fy c c k k + 4 + + 4= 0 represent + + + + = 0: g = k so g = k To represent a circle, f = 4 so f = = + 4. g f c ( ) k k k + > 0 + 4 + 4 > 0 k + 8> 0 k < 8. 4 Intersection of a Line and a Circle A A straight line and circle can have two, one or no points of intersection: two intersections one intersection no intersections If a line and a circle only touch at one point, then the line is a tangent to the circle at that point. To find out how many times a line and circle meet, we can use substitution. EXAMPLES 1. Find the points where the line with equation y = 3x intersects the circle with equation x + y = 0. x + y = 0 x + ( 3x) = 0 Remember x + 9x = 0 ab m a m b m. 10x = 0 x = x = ± x = x = y = 3( ) = 3 y = 3( ) = 3 So the circle and the line meet at (,3 ) and (, 3 ). hsn.uk.net 5
. Find the points where the line with equation y = x + 6 and circle with equation x + y + x + y 8= 0 intersect. Substitute y = x + 6 into the equation of the circle: x + ( x + 6) + x + ( x + 6) 8= 0 x + ( x + 6)( x + 6) + x + 4x + 1 8= 0 + 4 + 4 + 36 + + 4 + 1 8 = 0 x x x x x x + = 0 x = = ( ) + 6= x + x + = 5 30 40 0 ( x x ) 5 + 6 + 8 = 0 ( x )( x ) + + 4 = 0 x + 4= 0 x = 4 = 4 + 6 =. y y ( ) So the line and circle meet at (, ) and ( 4, ). 5 Tangents to Circles A As we have seen, a line is a tangent if it intersects the circle at only one point. To show that a line is a tangent to a circle, the equation of the line can be substituted into the equation of the circle, and solved there should only be one solution. EXAMPLE Show that the line with equation x + y = 4 is a tangent to the circle with equation x + y + 6x + y = 0. Substitute y using the equation of the straight line: + + 6 + = 0 x y x y x ( x) x ( x) ( )( ) ( ) + 4 + 6 + 4 = 0 + 4 4 + 6 + 4 = 0 x x x x x + 16 8 + + 6 + 8 = 0 x x x x x x 4 0 ( x x ) + 1 = 0 x x + = x + 1 = 0. hsn.uk.net 6
Then (i) factorise or (ii) use the discriminant x x 1= 0 x + 1= 0 ( x 1)( x 1) = 0 x = 1 x 1= 0 x = 1. Since the solutions are equal, the line is a tangent to the circle. x x + 1= 0 a = 1 b 4ac b = c = 1 = = 4 4 = 0. Since b 4ac = 0, the line is a tangent to the circle. ( ) 41 ( 1) Note If the point of contact is required then method (i) is more efficient. To find the point, substitute the value found for x into the equation of the line (or circle) to calculate the corresponding y-coordinate. 6 Equations of Tangents to Circles A If the point of contact between a circle and a tangent is known, then the equation of the tangent can be calculated. If a line is a tangent to a circle, then a radius will meet the tangent at right angles. The gradient of this radius can be calculated, since the centre and point of contact are known. Using mradius mtangent = 1, the gradient of the tangent can be found. The equation can then be found using y b = m( x a), since the point is known, and the gradient has just been calculated. hsn.uk.net 7
EXAMPLE Show that A ( 1, 3 ) lies on the circle equation of the tangent at A. Substitute point into equation of circle: x + y + 6x + y x y x y + + 6 + = 0 and find the = 1 + 3 + 6( 1) + ( 3) = 1+ 9+ 6+ 6 = 0. Since this satisfies the equation of the circle, the point must lie on the circle. Find the gradient of the radius from ( 3, 1) to ( 1, 3 ): y y1 mradius = x x1 3+ 1 = 1 + 3 = 1. So m = 1 since m m = 1. tangent radius tangent Find equation of tangent using point ( 1, 3 ) and gradient m = 1: y b = m( x a) y 3= ( x 1) y 3= x + 1 y = x + 4 x + y 4 = 0. Therefore the equation of the tangent to the circle at A is x + y 4= 0. hsn.uk.net 8
7 Intersection of Circles A Consider two circles with radii r 1 and r with r > r. 1 Let d be the distance between the centres of the two circles. r 1 d r d > r1+ r d The circles do not touch. d = r1+ r r1 r < d < r1+ r d d The circles touch externally. The circles meet at two distinct points. Note Don t try to memorise this, just try to understand why each one is true. d = r1 r d The circles touch internally. d < r1 r d The circles do not touch. EXAMPLES 1. Circle P has centre ( 4, 1) x y x y and radius units, circle Q has equation + + 6 + 1= 0. Show that the circles P and Q do not touch. To find the centre and radius of Q: Compare with x + y + gx + fy + c = 0: g = so g = 1 Centre is ( g, f ) Radius rq = g + f c f = 6 so f = 3 = ( 1, 3 ). = 1+ 9 1 c = 1. = 9 = 3 units. hsn.uk.net 9
We know P has centre ( 4, 1) and radius r P = units. So the distance between the centres d = ( 1+ 4) + ( 3+ 1) ( ) = + 5 = 9 = 5. 39 units (to d.p.). Since r P + r Q = 3+ = 5< d, the circles P and Q do not touch.. Circle R has equation equation ( x ) ( y ) externally. x y x y + 4 4= 0, and circle S has 4 + 6 = 4. Show that the circles R and S touch To find the centre and radius of R: Compare with g = so g = 1 f = 4 so f = x y gx fy c c = 4. + + + + = 0: Centre is, To find the centre and radius of S: Compare with ( ) ( ) r a = 4 b = 6 = 4 so r =. ( g f ) = ( 1, ). x a y b r + =. Centre is, = ( ab) ( ) 4, 6. Radius rr = g + f c S = ( 1) + ( ) + 4 = 9 = 3 units. Radius r = units. So the distance between the centres d = ( 1 4) + ( 6) ( 3) ( 4) = + 5 = 5 units. Since r R + r S = 3+ = 5= d, the circles R and S touch externally. = hsn.uk.net 10
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CIRCLE ANSWERS 16
The Circle 1. Write down the equation of each circle below (a) Centre the Origin, radius 4 (b) Centre the Origin, radius 6 (c) Centre (-1,4), radius 5 (d) Centre (-,-5), radius 10. Write down the centre and radius of each circle below (a) x + y = 5 (b) x + y = 1 (c) (x 3) + (y ) = 36 (d) (x + 1) + (y 4) = 10 (e) x + y 10x 6y = 0 (f) x + y + 6x + 4y + 4 = 0 3. (a) The point (a,5) lies on the circle with equation x + y = 74. Find two values for a. (b) The point (3,c) lies on the circle x + y 4x + 6y + 1 = 0. Find c. 4. The lines x = -, x = 10, y = -5 and y = 7 are tangents to a circle. Find the equation of this circle. y 5. The circle shown has centre (4,7) and passes through the origin. Find its equation..(4,7) x 6. The diagram shows the circle with equation (x 4) + (y + 5) = 40. P(,1) Find the equation of the tangent to this circle at the point P(,1). 7. The diagram shows the circle x + y 6x 4y + 8 =0. Find the equation of the tangent to this circle at the point A(5,1). A(5,1) 17
8. Find the equation of the tangent to the circle x + y 10y 43 = 0 at the point (,-3). 9. Find the points of intersection of the line y = x + 8 and the circle with equation x + y + 4x + y 0 = 0. 10. Find the points of intersection of the circle x + y x 4y + 1 = 0 and the line x + y = 1. y 11. The straight line y = x cuts the circle x + y 6x y 4 = 0 at A and B. A y = x (a) Find the coordinates of A and B. (b) Find the equation of the circle which has AB as diameter. B x x + y 6x y 4 = 0 1. Show that the line y = -3x 10 is a tangent to the circle x + y 8x + 4y 0 = 0, and find the point of contact. 13. The circle,centre C, has equation x + y 4x + 6y 1 = 0. y P. (a) Find the equation of the tangent at the point A(5,1) on this circle. A The line through P(1,4) at right angles to this tangent has equation 4x 3y + 8 = 0. x (b) Show that this line is also a tangent to the circle..c 14. In the diagram, y The circle, centre A, has equation x + y + x 8y 8 = 0. The circle, centre B, has equation x +y x + 10y + 11 = 0. The line PQ passes through A and B. Calculate the length of the line PQ. P.A.B x Q 18
y 15. In the diagram opposite, the centres A, B and C are collinear. The equations of the outer circles are (x + 1) + (y + 15) = 5 and (x 4) + (y 1) = 100. Find the equation of the central circle. C.B x A 19
The Circle 1. Find the equation of the circle centre (-4,7) which has the x-axis as a tangent.. Find the equation of the circle which has the lines x = -4, x = 8, y = - and y = 10 as tangents. 3. A circle has equation x + y 4x 8y 5 = 0. Write down the equation of the tangent to this circle at the point (-3,4). 4. A circle has equation (x + 5) + (y 1) = 16. Write down the equation of the tangent to this circle at the point A(-5,-3). 5. A circle has equation x + y + 6x + 4 =0. Find the equation of the tangent to this circle at the point P(-5,-1). P(-5,-1) 6. Find the equation of the tangent to the circle x + y 8x + y 3 = 0 at the point A(,3). A(,3) 7. A is the point (-4,) and B is (6,-4). Find the equation of the circle which has AB as a diameter. A(-4,) 8. P is the point (-5,3) and Q is (5,-1). Find the equation of the circle which has PQ as diameter. B(6,-4) 0
9. Two congruent circles with centres A and B touch at G. The equations of the circles are x + y + 8x 4y 5 = 0 and x + y 4x 0y + 79 = 0 (a) Find the coordinates of G. (b) Find the length of AB.. A G. B 10. Two circles have equations (x + 1) + (y + 3) = 0 and x + y 10x 18y + 6 = 0 (a) Write down the centre and radius of each circle. (b) Show that the circles touch at a single point. (c) Find P, the point of contact of the circles. 11. Two circles have equations x + y + 4x + 16y 60 = 0 and x + y 8x + 4y + 1 = 0 Show that these circles touch at a single point. 1. Three circles touch externally as shown. The centres of the circles are collinear (x + ) + (y 8) = 9 and the equations of the two smaller circles are (x + ) + (y 8) = 9 and x + y 0x + 16y + 155 = 0 Find the equation of the larger circle. x + y 0x + 16y + 155 = 0 1
13. The circle x + y + 4x 7y 8 = 0 cuts the y-axis at two points. Find the coordinates of these points. 14. The circle x + y x + 10y 4 = 0 cuts the x-axis at the points A and B. Find the length of AB. 15. (a) A circle has equation (x + 3) + (y 6) = 61. Find the equation of the tangent to this circle at the point A(3,3). (b) Show that this tangent is also a tangent to the circle with equation x + y + 6x 7y 10 = 0 and find the point of contact. A(3,3) 16. Show that the line y = -3x 10 is a tangent to the circle with equation x + y 8x + 4y 0 = 0 and find the point of contact. 17. (a) Find the equation of the tangent to the curve y = x 3 4x 7x + 1 at the point where x =. (b) Show that this tangent is also a tangent to the circle x + y 6x + y + 10 = 0 and find the point of contact. 18. Show that the line y = x + 1 does not intersect the circle with equation x + y x + 4y + 1 =0. 19. For what range of values of p does the equation x + y + px + py + 6p + 8 = 0 represent a circle. 0. For what range of values of k does the equation x + y kx + 4ky + 4 k = 0 represent a circle. 1. (a) A circle has centre (a,0), a > 0 and radius 4 units. Write down the equation of this circle. (b) Show that if y = x is a tangent to this circle then a = 4. (a,0) 4
y. The diagram shows six identical circles. Circle A has equation x + y 6x 6y + 9 = 0. (a) Write down the equation of circle F. (b) Find the point of contact between the the circles C and D. B D F A C E x 3. (a) Find the equation of AB, the perpendicular bisector of the line joining the points P(-3,1) and Q(1,9). (b) C is the centre of a circle passing through P and Q. Given that QC is parallel to the y-axis, determine the equation of the circle. (c) The tangents at P and Q intersect at T. P(-3,1) A y Q(1,9). C Write down (i) the equation of the tangent at Q (ii) the coordinates of T. B x 4. The diagram shows a tangent kite ABCD and a circle centre C. A is the point (-8,0) and B is (4,9). The radius CD is parallel to the y-axis. y B(4,9) (a) Find the coordinates of D and write down the equation of CD. (b) Find the equation of the line BC. C (c) Find the coordinates of C and hence determine the equation of the circle. A(-8,0) D x 3
Higher Mathematics Circles 1.Thelinewithequationy =xintersectsthecirclewithequationx +y =5at thepointsjandk. Whatarethex-coordinatesofJandK? A. x J =1,x K = 1 B. x J =,x K = C. x J =1,x K = D. x J = 1,x K = [SQA]. Find the equation of the tangent at the point (3, 4) on the circle x +y +x 4y 15 =0. 4 [SQA] 3. Explainwhytheequationx +y +x +3y +5 =0doesnotrepresentacircle. [SQA] 4. FindtheequationofthecirclewhichhasP(, 1)andQ(4,5)astheendpoints of a diameter. 3 [SQA] 5. [SQA] 6. hsn.uk.net Page 1 Questions marked [SQA] c SQA All others c Higher Still Notes 4
Higher Mathematics [SQA] 7. Forwhatrangeofvaluesofkdoestheequationx +y +4kx ky k =0 represent a circle? 5 [SQA] 8. [SQA] 9. 10.ThepointP(, 3)liesonthecircle with centre C as shown.the gradient ofcpis.whatistheequationof thetangentatp? C y O x P(, 3) A. y +3 = (x ) B. y 3 = (x +) C. y +3 = 1 (x ) D. y 3 = 1 (x +) hsn.uk.net Page Questions marked [SQA] c SQA All others c Higher Still Notes 5
Higher Mathematics [SQA] 11. Theliney = 1isatangenttoacirclewhichpassesthrough (0,0)and (6,0). Find the equation of this circle. 6 [SQA] 1. CirclePhasequationx +y 8x 10y +9 =0. CircleQhascentre (, 1) andradius. (a) (i)showthattheradiusofcirclepis4. (ii)henceshowthatcirclespandqtouch. 4 (b)findtheequationofthetangenttothecircleqatthepoint ( 4,1). 3 (c)thetangentin(b)intersectscirclepintwopoints.findthex-coordinatesof thepointsofintersection,expressingyouanswersintheforma ±b 3. 3 [SQA] 13. (a)showthatthepointp(5,10)liesoncirclec 1 withequation (x +1) + (y ) =100. 1 (b)pqisadiameterofthiscircleas y showninthediagram. Findthe equationofthetangentatq. P(5, 10) 5 O x Q (c)twocircles,c andc 3,touchcircleC 1 atq. TheradiusofeachofthesecirclesistwicetheradiusofcircleC 1. FindtheequationsofcirclesC andc 3. 4 hsn.uk.net Page 3 Questions marked [SQA] c SQA All others c Higher Still Notes 6
Higher Mathematics [SQA] 14. [SQA] 15. hsn.uk.net Page 4 Questions marked [SQA] c SQA All others c Higher Still Notes 7
Higher Mathematics [SQA] 16. [SQA] 17. [END OF QUESTIONS] hsn.uk.net Page 5 Questions marked [SQA] c SQA All others c Higher Still Notes 8
Higher Mathematics Circle Homework 9
St Andrew s Academy Maths Dept 016-17 Higher 30