SYSTEM OF CIRCLES OBJECTIVES (a) Touch each other internally (b) Touch each other externally

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1 SYSTEM OF CIRCLES OBJECTIVES. A circle passes through (0, 0) and (, 0) and touches the circle x + y = 9, then the centre of circle is (a) (c) 3,, (b) (d) 3,, ±. The equation of the circle having its centre on the line x + y 3 and passing through the points of intersection of the circles x + y x 4y + and x + y 4 x y + 4, is (a) x + y 6 x + 7 (b) x + y 3y + 4 (c) x + y x y + (d) x + y + x 4y The point of contact of the given circles x + y 6 x 6y + 0 and x + y =, is (a) (0, 0) (b) (, ) (c) (, ) (d) (, ) 4. Circles x + y x 4y and x + y 8y 4 (a) Touch each other internally (b) Touch each other externally (c) Cuts each other at two points (d) None of these 5. If the circles x + y = a and x + y gx + g b touch each other externally, then (a) g = ab (b) g = a + b (c) (d) g = a + b g = ab 6. The radical centre of the circles x + y + 4 x + 6y = 9, x + y = 9 and x + y x y = 5 will be (a) (, ) (b) (, ) (c) (, ) (d) (0, ) 7. The equation of a circle passing through points of intersection of the circles x + y + 3 x 3y and x + y + 4 x 7y 5 and point (, ) is (a) 4 x + 4y 30 x 0 y 5 (b) 4 x + 4y + 30 x 3y 5 (c) 4 x + 4y 7 x 0 y + 5 (d) None of these

2 8. The equation of the circle which intersects circles + y + x + y + 3 x + y 7 x 8y 9 at right angle, will be (a) x + y 4 x 4y 3 (b) 3( x + y ) + 4 x 4y 3 x, x + y + x + 4y + 5 and (c) x + y + 4 x + 4y 3 (d) 3( x + y ) + 4( x + y) 3 9. Two given circles x + y + ax + by + c and x + y + dx + ey + f will intersect each other orthogonally, only when (a) a b + c = d + e + f + (b) ad + be = c + f (c) ad be = c + f + (d) ad + be = c + f 0. The locus of centre of a circle passing through (a, b) and cuts orthogonally to circle x + y = p, is (a) + by ( a + b + p ) ax (b) ax + by ( a b + p ) (c) x + y 3ax 4by + ( a + b p ) (d) x + y ax 3by + ( a b p ). The equation of the circle through the point of intersection of the circles x and (3, 3) is x + y 8 x y + 7, + y 4 x + 0y + 8 (a) 3 x + 3y 56 x + 38 y + 68 (b) 3 x + 3y + 56 x + 38 y + 68 (c) x + y + 56 x + 38 y + 68 (d) None of these. The equation of line passing through the points of intersection of the circles 3x + 3y x + y 9 and x + y + 6 x + y 5 (a) 0 x 3y 8 (b) 0 x + 3y 8 (c) 0 x + 3y + 8 (d) None of these 3. The locus of the centres of the circles which touch externally the circles x + y = a and +, will be x y = 4ax (a) x 4y 4 ax + 9a (b) x + 4y 4 ax + 9a (c) x 4y + 4 ax + 9a (d) x + 4y + 4 ax + 9a 4. If the circles of same radius a and centers at (, 3) and (5, 6) cut orthogonally, then a =, is (a) (b) (c) 3 (d) 4

3 5. Two circles S = x + y + g x + f y + c 0 and S = x + y + g x + f y + c 0 cut each other = orthogonally, then (a) g g + f f = c + c (b) g g f f = c + c (c) gg + f f = c c (d) gg f f = c c 6. Circles x + y + gx + fy and y x + + g ' x + f' y = = 0 touch externally, if (a) f ' g = g' f (b) fg = f' g' (c) f ' g' + fg (d) f ' g + g' f 7. Consider the circles + (y ) = (a) These circles touch each other x 9, ( x ) + y = 5 (b) One of these circles lies entirely inside the other (c) Each of these circles lies outside the other (d) They intersect in two points. They are such that 8. One of the limit point of the coaxial system of circles containing + y 6 x 6y + 4 x + y x 4 y + 3 is (a) (,) (b) (, ) (c) (,) (d) (, ) x, 9. The equation of circle which passes through the point (,) and intersect the given circles x orthogonally, is x + y + x + 4y + 6 and + y + 4 x + 6y + (a) x + y + 6 x + y + (b) x + y 6 x y (c) x + y 6 x + y + (d) None of these 0. Locus of the point, the difference of the squares of lengths of tangents drawn from which to two given circles is constant, is (a) Circle (c) Straight line (b) Parabola (d) None of these. The locus of centre of the circle which cuts the circles x + y + g x + f y + c and x + y + g x + fy + c (a) An ellipse (c) A conic orthogonally is (b) The radical axis of the given circles (d) Another circle. P, Q and R are the centres and r, r, r3 are the radii respectively of three co-axial circles, then QRr + RP r + is equal to PQr3 (a) PQ. QR. RP (b) RP PQ. QR. (c) PQ. QR. RP (d) None of these

4 o 3. If the chord y = mx + of the circle x + y = subtends an angle of measure 45 at the major segment of the circle then value of m is (a) (b) (c) (d) None of these 4. The radical axis of two circles and the line joining their centres are (a) Parallel (c) Neither parallel, nor perpendicular (b) Perpendicular (d) Intersecting, but not fully perpendicular 5. Any circle through the point of intersection of the lines x + 3 y = and 3 x y = if intersects these lines at points P and Q, then the angle subtended by the arc PQ at its centre is (a) (c) o o 80 (b) 90 o 0 (d) Depends on centre and radius 6. If the straight line y = mx is outside the circle x + y 0y + 90, then (a) m > 3 (b) m < 3 (c) m > 3 (d) m < 3 7. The points of intersection of circles x y = ax + are + and x y = by (a) (0, 0), (a, b) (b)(0, 0),, + + a b a b ab ba (c) (0, 0), a + b a + b, a b (d)none of these 8. If circles x + y + ax + c and x + y + by + c touch each other, then (a) a = b c + (b) + = a b c a b (c) + = c (d) + a b = c 9. If two circles ( x ) + ( y 3) = r and x + y 8 x + y + 8 intersect in two distinct points, then (a) < r < 8 (b) r = (c) r < (d) r > 30. The circle x + y + gx + fy + c bisects the circumference of the circle + y + g' x + f' y + c' if x, (a) (c) g'( g g') + f'( f f ') = c c' (b) g' ( g g') + f '( f f ') = c c' f( g g') + g( f f') = c c' (d) None of these

5 3. The locus of the centre of a circle which cuts orthogonally the circle x + y 0 x + 4 and which touches x = is (a) y = 6 x + 4 (b) x = 6y (c) x = 6 y + 4 (d) y = 6 x 3. The equation of a circle that intersects the circle x + y + 4 x + 6y + orthogonally and whose centre is (0, ) is (a) x + y 4y 6 (b) x + y + 4y 4 (c) x + y + 4y + 4 (d) x + y 4y The equation of the circle which passes through the intersection of x + y + 3 x 3y and x + y + 4 x 7y 5 and whose centre lies on 3 x + 30 y is (a) x + y + 30 x 3y 5 (b) 4 x + 4y + 30 x 3y 5 (c) x + y + 30 x 3y 5 (d) x + y + 30 x 3y If the circle x + y + 6x y + k bisects the circumference of the circle x + y + x 6y 5, then k = (a) (b) (c) 3 (d) If P is a point such that the ratio of the squares of the lengths of the tangents from P to the circles x + y + x 4y 0 and + y 4 x + y 44 (a) (7, 8) (b) ( 7, 8) (c) (7, 8) (d) ( 7, 8) x is : 3, then the locus of P is a circle with centre 36. If d is the distance between the centres of two circles, r,r are their radii and d = r + r, then (a) The circles touch each other externally (b) The circles touch each other internally (c) The circles cut each other (d) The circles are disjoint 37. If the circles x + y = 4, x + y 0 x + λ touch externally, then λ is equal to (a) 6 (b) 9 (c) 6 (d) 5

6 38. If the circles x + y + ax + cy + a and x + y 3ax + dy intersect in two distinct points P and Q then the line 5 x + by a passes through P and Q for (a) Infinitely many values of a (b) Exactly two values of a (c) Exactly one value of a (d) No value of a 39. The radical axis of the pair of circle x + y = 44 and x + y 5 x + y is (a) 5 y x (b) 3 x y = (c) 5 x 4 y = 48 (d) None of these 40. The equation of radical axis of the circles x + y + x y + and 3x + 3y 4 x, is (a) x + y 5 x + y 4 (b) 7 x 3y + 8 (c) 5 x y + 4 (d) None of these 4. If the circles x + y ax + c and x + y + by + λ intersect orthogonally, then the value of λ is (a) c (b) c (c) 0 (d) None of these 4. A circle touches the x-axis and also touches the circle with centre at (0, 3) and radius. The locus of the centre of the circle is (a) A hyperbola (b) A parabola (c) An ellipse (d) A circle 43. The number of common tangents to the circles + y = 4 (a) 0 (b) (c) 3 (d) 4 x and x + y 6 x 8y = 4 is

7 SYSTEM OF CIRCLES HINTS AND SOLUTIONS. (d) Radius of the circle,k 3 r = (,0) 9 + k = k = ± 4 4 Hence centre is, ±. (a) Required circle will be + S 0 S λ, λ = i.e., x + y x 4y + + λ ( x + y 4 x y + 4) ( + λ) ( + λ) + 4λ x + y x y + + λ + λ + λ Its centre + λ + λ, + λ + λ lies on x + y 3 λ =. The circle is x + y 6 x (b) x + y 6 x 6y + 0.(i) x + y =.(ii) By verification ans.(,) 4. (a) C, ), C (0, 4), R = 5, R 5 ( = C C 5 and C C = R = R Hence circles touch internally.

8 5. (d) According to the condition, ( g 0) + (0 0) = a + b g = a + b. 6. (a) Radical axes are 4 x + 6y or + 3y = 5 x + y = 4 or + y = x.(i) x.(ii) Point of intersection of (i) and (ii) is (, ). 7. (b) Required equation is ( x + y + 3 x 3y) + λ (x + y + 4 x 7y 5) Which passes through (, ), so λ =. Hence required equation is 4 x + 4y + 30 x 3y (d) Verification. ad be + 9. (c) From Condition for orthogonal intersection, = c + f 0. (a) Let equation of circle be x + y + gx + fy + c with x + y = p cutting orthogonally, we get + 0 = +c or p 0 p a + b + ga + fb + p Or c = and passes through (a, b), we get ax + by ( a + b + p ) Required locus as centre ( g, f) is changed to (x, y).. (a) Equation of circle is ( x + y 8 x y + 7) + λ ( x + y 4 x + 0 y + 8) Also point ( 3, 3) lies on the above equation. λ = 7 6 Hence required equation is 3 x + 3y 56 x + 38 y (a) Common chord = S S 0 x 3y 8.

9 3. (a) Let C ( h, k), radius = r r a A B a (a,0) S =x +y a S =x +y 4ax Co-ordinates of ah ak A, a + r a + r Co-ordinates of ar + ah ak B, a + r a + r Putting co-ordinates of A and B in S, S respectively and eliminating r, we get the locus x 4y 4 ax + 9a. 4. (c) ( C ) = r + r a = 8 a 3 C. = 5. (a) concept. 6. (a) OA + O A = OO' g + f + f' + g' = ( g' g) + ( f' f) g + f + f' + g' + g + f f' + g' = ( g' g) + ( f' f) g + f f' + g' = ( gg' + ff' ) (0,0) O O g f' + f g' = gg' ff'. ( gf ' fg' ) gf' = fg'. 7. (b) Centres and radii of the given circles are Centres: C 0, ), (, 0) ; Radii : =, r 5 ( C r. 3 = Clearly, C C = < ( r ). Therefore one circle lies entirely inside the other. r 8. (a) The equation of radical axis is S S 0 i.e., 4 + y = x. The equation of circle of co-axial system can be taken as ( + y 6 x 6y + 4) + (4 x + y ) Or x + y (6 4λ) x (6 λ) y + (4 λ).(i) x λ

10 Whose centre is C ( 3 λ, 3 λ) and radius is r = (3 λ) + (3 λ) (4 λ) If r, then we get λ = or / 5 7. Putting the co-ordinates of C, the limit points are (, ) and 8, 5 5. One of these limit points is given in (a). 9. (c) Let equation of circle be + y + gx + fy + c x. As it intersects orthogonally the given circles, we have g + 4 f = 6 + c and 4 g + 6 f = + c. As it passes through (, ), we have g + f = c From these, we get x + y 6 x + y +. g, f and c as 8, 6, respectively and hence equation of circle as 0. (c) t t = k S S = k st degree equation which represents a straight line parallel to radical axis S 0 S. =. (b) Radical axis of the given circles.. (b) As g g g ) = ( g g )( g g )( g ) Σ. ( g 3. (c) Given circle is x + y = C (0,0) and radius = and chord is y = mx + CP cos 45 o = CR 45 o C (0, 0) 45 o 45 o P R y = mx + CP = Perpendicular distance from (0,0) to chord y = mx + CP = m + (CR = radius = ) / cos 45 = m + = m + m + = m = ±. 4. (b) We know that the radical axis of two circles is the locus of a point which moves in such a way that the lengths of the tangents drawn from it to the two circles are equal. It is a line perpendicular to the line joining the centres of two circles.

11 5. (a) Let the point of intersection of two lines is A. x + 3y = A θ θ 3 x y = P The angle subtended by PQ on centre C = Two times the angle subtended by PQ on point A. For + 3 y = x, = 3 m and For 3 y =, x m 3 = o m m = 3 =, A = 90 3 The angle subtended by arc PQ at its centre = 90 = 80 o o 6. (d) If the straight line y = mx is outside the given circle then distance from centre of circle > 0 radius of circle > 0 + m ( + m ) < 0 m < 9 m < (b) Given circles are x + y = ax..(i) and +..(ii) x y = by (i) (ii) 0 ( ax by ) a = y = x b From (i), a x + x = ax a x b + x a b ab x, For a + b x, y and for ab a b x =, y = a + b a + b The points of intersection are (0, 0) and ab a + b a b, a + b. 8. (d) (, 0); C C (0, ); R ( a ); a R ( b c) b c C = a b C + Since they touch each other, therefore

12 a c + b c = a + b a b b c a c Multiply by, a b c we get + =. a b c 9. (a) When two circles intersect each other, then Difference between their radii<distance between centers r 3 < 5 r = 8...(i) Sum of their radii > Distance between centres r + 3 > 5 r > Hence by (i) and (ii), < r < 8....(ii) 30. (a) Common chord S S 0 passes through the centre of 0 3. (d) Let the circle be = x + y + gx + fy + c...(i) It cuts the circle x + y 0 x + 4 orthogonally S. = ( 0 g + 0 f) = c = c + 4 g...(ii) Circle (i) touches the line x =, x + 0 y g + 0 = g + f + 0 c 4...(iii) ( g + ) = g + f c g + 4 = f c Eliminating c from (ii) and (iii), we get 6 g 4 = f 4 f + 6 g Hence the locus of ( g, f) is 6 x y. 3. (d) In circle, x + y + 4 x + 6y + g = 7, f = 3, c = Centre of circle ( g, f) = (0, ), (Given) For orthogonally intersection, gg ' + ff' = c + c' 0 = + c ' c' = 4 Put the values, in equation + y + g' x + f' x + c' x.

13 x + y + 0 4y 4 x + y 4y (b) The equation of required circle is + S 0 S λ. = 7 5 x ( + λ) + y ( + λ) + x( + 3λ) y + 3λ Centre = ( + 3λ), 7 + 3λ Centre lies on 3 x + 30 y 7 + 3λ + 3λ λ =. Hence the equation of required circle is 4 x + 4y + 30 x 3y (d) g ( g g) + f( f f ) = c c () (3 ) + ( 3) ( + 3) = k = k + 5 or 8 = k + 5 k = (b) x x + y + y + x 4y 0 4 x + y 44 = 3 x, Centre = 7,8) + y + 4 x 6 y + 8 (. 36. (a) r r d Clearly, circles touch each other externally. 37. (a) Circles x + y = 4, x + y 0x + λ touch externally C C = r + r (0, 0) = C and C ) r = 5 + r and λ = (5,0 (5 0) + 0 = λ 5 = 5 + λ 3 = 5 + λ 9 = 5 + λ λ = (d) Equation of line PQ (i.e., common chord) is 5 ax + ( c d) y + a +...(i) Also given equation of line PQ is

14 5 + by a x...(ii) 5a c d a + a + = As = a 5 b a a Therefore = ; a + a + Therefore no real value of a exists, (as D<0). 39. (c)the radical axis of the circle S and ( x + y 44 ) ( x + y 5 x + y) 5 x y 44 5 x 4 y = (b) Radical axis is S S S x + y + x y + 4 S = x + y x S S = x y Or 7 3y + 8 x. 4. (d) Condition for circle intersects orthogonally, ( g = c + c g + f f ) c 0 = c + λ λ = 4. (b) Let centre ( h, k); As C C = r + r,(given) S is S S 0 = ( h 0) + ( k 3) = k + h = 5(k ) Hence locus, x = 5(y ), which is parabola. 43. (b) Circles x + y = S, S ( x 3) + ( y 4) = 7 Centres 0), C = (0, C = (3, 4) and radii =, r 7 r C C 5, r r 5 = = = i.e., circles touch internally. Hence there is only one common tangent.

15 SYSTEM OF CIRCLES PRACTICE EXERCISE. The distance from (, ) to the R.A. of the circles x + y +6x-6=0, x +y -x+6y- 6=0 is ) ) 3) 3 4) 7/5. The radical centre of the circles x + y = 9, x + y +4x+6y-9=0, x +y -x-y- 5=0 is ) (0, 0) ) (0, ) 3) (, 0) 4) (, ) 3. The centre of the circle orthogonal to x +y +4y+, x +y +6x+y 8, x +y 4x-4y+37=0 is ) (,) ) (,) 3) (3,3) 4) (0,0) 4. The radius of the circle which cuts orthogonally circles x + y - 4x - y + 6, x + y - x + 6y, x + y - x + y + 30 is ) ) 3 3) 6 4) 5. The equation of the circle passing through (0, 0) and cutting orthogonally the circles x +y +6x-5, x + y - 8y - 0 is ) x + y - 0x + 5y ) x + y - 0x + 5y 3) x + y + 0x + 5y 4) x + y + 0x + 5y 6. The equation of the circle passing through the origin and the points of intersection of x +y = 4, x + y - x - 4y + 4 is ) x + y + x + y ) x + y - x -y 3) x + y + x - y 4) x + y - x + y

16 7. The equation of the circle passing through the origin and the points of intersection of the two circles x + y - 4x - 6y - 3 and x + y +4x - y - 4 is ) x + y + 8x + 8y ) x + y - 8x - 8y =0 3) x + y -4x -9y 4) x + y + 4x + 9y 8. The equation of the circle passing through the points of intersection of x +y -x-=0, x + y + 5x - 6y + 4 and cutting orthogonally the circle x +y = 4 is ) x + y - 5x - 6y + 4 ) x + y + 5x + 6y + 4 3) x + y - 5x + 6y - 4 4) x + y + 5x - 6y The equation of the circle whose diameter is the common chord of the circles x +y +x+3y+, x + y + 4x + 3y + is )x + y + x + 3y + ) x + y + x + 6y + 3)x + y + x + 6y + 4) x + y + x + 6y +3 =0 0. The line x+3y= cuts the circle x + y = 4 in A and B. The equation of the circle on AB as diameter is ) 3x + 3y - 4x - 6y - 50 ) 3x + 3y - 4x - 6y - 5 3) 3x + 3y + 4x + 6y ) 3(x +y )+4x+6y+5=0. 3 circles with centres A, B, C intersect orthogonally the radical centre of the three circles is ) In centre of ABC ) Orthocentre of ABC 3) Circumcentre of ABC 4) Centroid of ABC. The centres of two circles are (a, c), (b, c) and their common chord is the y - axis. If the radius of one circle is r, the radius of the other circle is ) r + b a ) r + b a 3) r + b + a 4) r + a b 3. The equations of four circles are (x±a) + (y±a) = a. The radius of a circle touching all the four circles externally is ) a ) ( + )a 3) ( )a 4) ( + )a

17 4. The circle passing through the points of intersection of the circles x +y - 3x-6y+8, x +y - x - 4y+4=0 and touching the line x+y = 5 is ) x + y -x - y ) x + y = 4 3) x + y + 4 4) x + y + x+y=0 5. The locus of the centre of the circle of radius which rolls on the outside the circle x + y + 3x 6y- 9=0 is ) x + y + 3x 6y + 5 =0 ) x + y + 3x 6y 3 3) x + y + 3x 6y + 4) 4(x + y +3x-6y)+9 6. The equation to the circle whose diameter is the common chord of the circles (x - a) + y = a ; x + (y - b) = b is ) x + y - (ax+by) ) x + y - ab(ax+by)=0 3) x + y + ab(ax+by)=0 4) (a + b )(x + y ) - ab(bx + ay) 7. The equation of the circle passing through the points of intersection of x + y + x + 8y - 33, x + y = 5 and touching the x-axis is ) x + y + 6x + 4y + 9 ) x + y - 6x + 4y + 9 3) x + y - 6x - 4y + 9 4) None 8. x = is the common chord of two circles intersecting orthogonally. If the equation of one of circle is x + y = 4, then the equation of the other circle is ) x + y + 8x + 4 ) x +y + 8x - 4 3) x + y - 8x + 4 4) x + y + x - 6y + 6 =0 9. There are two circles whose equations are x + y = 9 and x + y 8x 6y + n, n Z. If the two circles have exactly two common tangents, then the number of possible values of n is ) ) 8 3) 9 4) 5

18 0. A rod PQ of length a slides with its ends on the axes. The locus of the circumcentre of OPQ is ) x + y = a ) x + y = 4a 3) x + y = 3a 4) x + y = a. The number of common tangents of the circles x + y + gx + 8 and x + y + µ y 8 is ) ) 3) 3 4) 4. The intercept on the line y = x by the circle x + y x is AB. Equation of the circle on AB as a diameter is ) x + y + x + y ) x + y x + y 3) x + y x y 4) x + y + x y 3. x + y + px + b =0 and x + y + qx + b =0 are two circles. If one circle entirely lies in another circle, then ) pq < 0, b > 0 ) pq < 0, b < 0 3) pq>0, b>0 4) pq>0, b<0 SYSTEM OF CIRCLES PRACTICE TEST KEY

SYSTEM OF CIRCLES If d is the distance between the centers of two intersecting circles with radii r 1, r 2 and θ is the

SYSTEM OF CIRCLES If d is the distance between the centers of two intersecting circles with radii r 1, r 2 and θ is the SYSTEM OF CIRCLES Theorem: If d is the distance between the centers of two intersecting circles with radii r 1, r 2 and θ is the 2 2 2 d r1 r2 angle between the circles then cos θ =. 2r r 1 2 Proof: Let