Ei Exercise = Marked Questions may have for Revision Questions. PART -1: SUBJECTIVE QUESTIONS Section (A) Equation of circle, parametric equation, position of a point A-1. Find the equation of the circle that passes through the points (1, 0), (- 1, 0) and (0, 1). A-2. ABCDisa square in first quadrant whose side is a, taking AB and AD as axes, prove that the equation to the circle circumscribing the square is x? + y? = a(x + y). A.3. Find the equation to the circle which passes through the origin and cuts off intercepts equal fo 3 and 4 from the positive axes. ‘A-4. Find equation of circle which touches x & y axis & perpendicular distance of centre of circle from 3x + 4y + 11 = Ois 5. Given that circle lies in I" quadrant ‘A-5.x Find the equation to the circle which touches the axis of x at a distance 3 from the origin and intercepts a distance 6 on the axis of y. A-6. Find equation of circle whose cartesian equation are x = -3 +2 sin ,y=4+2 cos ® Aq. Find the values of p for which the power of a point (2, 5) is negative with respect to a circle x? + y? — 8x — 12y + p= O which neither touches the axes nor cuts them. Section (B) : : Line and circle, tangent, pair of tangent B-1, Find the points of intersection of the line x—y + 2= 0 and the circle 3x? + 3y*— 29x— 19y + 56 = 0. Also determine the length of the chord intercepted. B-2. Show that the line 7y — x= 5 touches the circle x? + y? ~ 5x + 5y = O and find the equation of the other parallel tangent. B3. Find the equation of the tangents to the circle x* + y* = 4 which make an angle of 60° with the positive x-axis in anticlockwise direction . B.4. Show that two tangents can be drawn from the point (9, 0) to the circle x? + y? = 16; also find the equation of the pair of tangents and the angle between them. BS, Ifthe length of the tangent from (f, g) to the circle x? + y? = 6 be twice the length of the tangent from (f, g) to the circle x? + y? + 3x + 3y = 0, then will f?+ 9? + 4f + 4g + 2=07 Section (C) : Normal, Director circle, chord of contact, chord with point C-1, Find the equation of the normal to the circle x? + y?= atthe point (1, 2) ©-2. Find the equation of the normal to the circle x? + y’ = 2x, which is parallel to the line x + 2 ©-3. Find the equation of director circle of the circle (x + 4)? + y? C-4.m Tangents are drawn to the circle x? + y* = 12 at the points where it is met by the circle x? + y?— 5x + 3y 2 = 0; find the point of intersection of these tangents. -5.m Tangents are drawn from the point (h,k) to the circle x? + y= a; prove that the area of the triangle 2 ak? a? formed by them and the straight ine joining their points of contact is 2° *K* =a") tak 6-6. Find the equation of the chord of the circle x* + y! + 6x + By + 9 = 0 whose middle point is (- 2, -3) Scanned with CamScannerCircle tas ‘Section (D) : Position of two circle, Orthogonality, Radical axis and radical centre Det. Find the equations to the common tangents of the circles x? + y? ~ 2x — 6y + 9 = Oand wet y+ Ox 2y+1=0 D-2, Show that the circles x? + y?~2x— 6y - 12 = O and x? + y? + 6x + 4y ~6 = 0 cut each other orthogonally. D-3.% Find the equation of the circle passing through the origin and cutting the circles x? + y? — 4x + Gy + 10 = 0 and x? + y? + 12y + 6 = 0 at right angles. D-4, Given the three circles x? + y? — 16x + 60 = 0, 3x?+ 3y?— 36x + 81 = 0 and x? + y?~ 16x ~ 12y + 84=0, find (1) the point from which the tangents to them are equal in length and (2) this length. Section (E) : Family of circles , Locus, Miscellaneous E41, Find the equation of the circle circumscribing the triangle formed by the lines x + y= 6, 2x +y = 4 and x+2y=5 E-2as If y = 2x is a chord of the circle x? + y? — 10x = 0, find the equation of a circle with this chord as diameter. PART - Il : ONLY ONE OPTION CORRECT TYPE Section (A) : Equation of circle, parametric equation, position of a point A-1, The radius of the circle passing through the points (1, 2), (5, 2) & (5, —2) is: (a) 52 (8) 2V5 (©) 3N2 p08 A2, The centres of the circles x? + y2 ~ 6x - 8y ~ 7 = 0 and x? + y2— 4x ~ 10y ~3 = 0 are the ends of the diameter of the circle x2 + y?— 5x — y +26 (B) x? + y2 + 5x 9y + 14=0 (C) x2 + y2 + 5x-y-14=0 (D) x2 + y2+ 5x4 y+ 14=0 A3. The circle described on the line joining the points (0, 1), (a,b) as diameter cuts the x-axis in points whose abscissa are roots of the equation: (A)e@tax+b=0 (B)x*-ax+b=0 — (C)x*+ax-b=0 —(D) x*-ax-b=0 A-4. The intercepts made by the circle x? + y?~ 5x ~ 13y - 14 = 0 on the x-axis and y-axis are respectively (A) 9, 13 (8) 5, 13 (C)9, 15 (D) none of these ‘A-6. Equation of line passing through mid point of intercepts made by circle x? + y? — 4x — 6y = 0 on co-ordinate axes is (A)3x+2y-12=0 (B)3x+y-6=0 — (C)3x+4y-12=0 (D)3x+2y-6=0 A-6.x% Two thin rods AB & CD of lengths 2a & 2b move along OX & OY respectively, when is the origin. The equation of the locus of the centre of the circle passing through the extremities of the two rods. (Aye tyre atebt (B)xt—yte arb? (C)xttyt=at—b? (OD) xt-y*= at + DF ‘A-T. _ LetAand B be two fixed points then the locus of a point C which moves so that (tanZBAC)(tan ZABC)=1, 0< 2BAC < 5,0< ZABC < (A) Circle (8) pair of straight ine (C)Apoint (0) Straightline Scanned with CamScannerAB. mek ey - ven ‘STATEMENT-1 : The length of intercept made by the circle x‘ + y* « STATEMENT-2 : x¢ + y?— ax ~ fly = Os a circle which passes through origin with centre (g + 8) and radius (®)__ STATEMENT-1 is true, STATEMENT.2 is true and STATEMENT2 Is correct explanation for ‘STATEMENT i (©) STATEMENT-1 is true, STATEMENT? is true and STATEMENT2 is not correct explanation for ‘STATEMENT-1 (©) STATEMENT-1 istrue, STATEMENT-2is false (0) _ STATEMENT-1 is false, STATEMENT-2 is true Section (B) : Line and circle, tangent, pair of tangent BA, B2. Bam B4. B55, Bs. B-7. BBm Bom Find the co-ordinates of a point p on line x + y =~ 13, nearest to the circle x? + y? + 4x + 6y-5=0 (A) (-6,-7) (B) (- 15, 2) (C) (-8,-6) (0) (-7,-6) The number of tangents that can be drawn from the point (8, 6) to the circle x? + y°~ 100 = O is (ayo (8)! (C2 (D) none of these Two lines through (2, 3) from which the circle x? + y2 = 25 intercepts chords of length 8 units have equations . (A) 2x + 3y = 13, x + 5y = 17 (B) y= 3, 12x + Sy = 39 (C) x= 2, 9x- My = 51 (D) y =, 12x + Sy = 39 The line 3x + 5y + 9 = O wrt the circle x2 + y?— 4x + 6y +5 =Ois (A) chord dividing circumference int : 3 ratio (B) diameter (C) tangent (D) outside line If one of the diameters of the circle x* + y? ~ 2x - 6y + 6 = 0 is a chord to the circle with centre (2, 1), then the radius of the circle is (A)3 (B)2 (C) 3/2 (0)1 The tangent lines to the circle x? + y? 6x + 4y = 12 which are parallel to the line 4x + 3y + 5 =O are given by: (A) 4x4 3y-7 = 0, 4x-+3y415=0 (B) 4x+ 3y-31 = 0, 4x +3y+19=0 (C) 4x+3y-17=0, 4x +3y +13 =0 (0) 4x+ 3y-31 = 0, 4x +3y-19=0 The condition so that the line (x + g) cos6 + (y + f) sin 8 is (A)g?+Ractk? (B)g’+Pact+k (C)g?+Ract+k? (D) g?+R=c+k The tangent to the circle x? + y? = 5 at the point (1, -2) also touches the circle x2 +y?- 8x + 6y +2050 at (A) (-2, 1) (B) (-3, 0) (C)(-1,-1) (D) (3, -1) The angle between the two tangents from the origin to the circle (x-7)*+ (y+ 1)? = is a tangent to x2 + y2+ 2gx + 2fy += 0 25 equals we Oy 5 (me B-10.m A point A(2, 1) Is outside the circle x*+ y* + 2gx+2fy+ c= 0 & AP, AQ are tangents to the circle, The Bat, equation of the circle circumscribing the triangle APQ is : (A) (x49) (x~2) + (y+ f) (y= 1) = 0 (B) (x+ 9) (x-2)- (y+f) (y-1) = (C) (xg) (x #2) + (yf) (y+1)=0 (D) (xg) (x2) + eae Aline segment through a point P cuts a given circle in 2 the length of tangent from points to the circle (ay7 (B) 25 (c)12 Points A& B, such that PA= 16 & PB=9, find (0)8 Scanned with CamScannerCircle N- B-12. The length of the tangent drawn from any point on the circle x? + y? + 2gx + 2fy + p = 0 to the circle B43. B-14.% The locus of the point of intersection of the tangents to the circle x? + y? C2. 3. Chaos C528 cs. C7 XP +? + 29x + 2fy +q = 0 is: (A) fa-p (8) yp-q (C) Javp (0) 2a + p The equation of the diameter of the circle (x — 2)? + (y + 1)?= 16 which bisects the chord cut off by the circle on the line x - 2y ~ 3 = is (A)x + 2y=0 (B)2x+y-3=0 — (C)3x+2y-4=0 (D) 3x-2y-4= ? at points whose parametric R angles differ by 3 is 2 2 wees @erys@ ary (Oye+y? 3 9 Section (C) : Normal, Director circle, chord of contact, chord with mid point CAL The equation of normal to the circle x? + y?— 4x + 4y - 17 = O which passes through (1, 1) is (A) 3x+y-4=0 (B)x-y=0 (C)x+y=0 (D) 3x-y-4=0 The normal at the point (3, 4) on a circle cuts the circle at the point (—1, -2). Then the equation of the circle is (A) x2 + y? + 2x - 2-13 = 0 (B) x2 + y2— 2x -2y-11=0 (C) x? + y2—2x + 2y +12=0 (D) x2 + y?— 2x -2y + 14 The co-ordinates of the middle point of the chord cut off on 2x ~ Sy + 18 = O by the circle xt + y2— 6x + 2y-54 = 0 are (A) (1, 4) (B) (2, 4) (C) (4, 1) (D) (1, 1) The locus of the mid point of a chord of the circle x? + y* = 4 which subtends a right angle at the origin is (A)xty=2 (B) x# + y* (C) et ye=2 (D)x+y=1 The chords of contact of the pair of tangents drawn from each point on the line 2x + y= 4 to the circle x? +y?= 4 pass through the point 44 (A) (1, 2) (B) (4) (C) (2, 4) (D) (4, 4) The locus of the centers of the circles such that the point (2, 3) is the mid point of the chord 5x + 2y = 16 is: (A) 2x-5y+11=0 (B)2x+Sy-11=0 (C) 2x+5y+11=0 — (D) 2x-Sy-11=0 Find the locus of the mid point of the chord of a circle x? + y*= 4 such that the segment intercepted by the chord on the curve x? ~ 2x - 2y = 0 subtends a right angle at the origin. (A) x2 + y?- 2x - 2y=0 (B) x+y? +2x-2y=0 (C) x+y? +2x+2y=0 (D) x+y? 2x + 2y=0 Section (D) : Position of two circle, Orthogonality, Radical axis and radical centre Da, D2. D3. Number of common tangents of the circles (x +2)*+(y-2)? = 49 and (x-2)? + (y + 1)?= 4s: OL) (B)1 (c)2 (03 The equation of the common tangent to the circle x? + y2 - 4x — 6y 12 = O and x? + y2 + 6x + 18y + 26 = 0 at their point of contact is (A) 12x + 5y +19 =0 (B) 5x + 42y + 19=0 (C) 5x -12y +19 =0 (D) 12x -5y +19 =0 Equation of the circle cutting orthogonally the three circles x? + y?- 2x + 3y — x2 + y2 + 5x — Sy +9 = O and x2 + y? + 7x 9y +29 = Os (A) x2 + y2— 16x - 18y -4 = (B) x2 + y?-7x+ Hy +6= (C) x2 + y2 + 2x - By +9 (D) x2 + y2 + 16x ~ 18y-4=0 Scanned with CamScannerCircle IN les is 7, and that of a common external tangent is D-4.m Ifthe length of a common internal tangent to two circl 11, then the product of the radii of the two circles is: yn (A) 18 (B) 20 (c) 16 (D) Section (E) : Family of circles , Locus, Miscellaneous E-1, The locus of the centre of the circle which bisects the circumferences of the circles xi+y? = 4 & x+y? 2x4 By +1 = 0 is: (A) a straight line (B) a circle (C)aparabola (D) pair of straight line E-2, Find the equation of the circle which passes through the point (1, 1) & which touches the circle x? + y? + 4x ~ Gy - 3 = 0 at the point (2, 3) on it. (A) x¥ +? +x - Gy + 3=0 (B) x? + y? +x -6y-3=0 (C)x? ty? +x 4 6y+3=0 (D) x + y?+x-3y+3=0 E3. Find the equation of circle touching the line 2x + 3y + 1 = 0 at (1, - 1) and cutting orthogonally the circle having line segment joining (0, 3) and (— 2, - 1) as diameter. (A) 2x? + 2y? = 10x- 5y #1 = 0 (B) 2x? + 2y? — 10x+ By + 1=0 (C) 2x? + 2y? - 10x- 5y-1=0 (D) 2x? + 2y? + 10x- Sy + 1=0 PART - Ill : MATCH THE COLUMN 4. Column -1 Column ~ I (A) Number of values of a for which the common chord 4 of the circles x? + y? = 8 and (x—a)? + y? = 8 subtends a right angle at the origin is (8) — Achord of the circle (x - 1)? + y?=4 lies along the @ 2 line y = 22 J (x — 1). The length of the chord is equal to (Cys. The number of circles touching all the three lines @ 0 3x+ Ty = 2, 21x + 49y = 5 and 9x + 2ty =O are (D)s._Ifradiiof the smallest and largest circle passing through ~) 1 the point (J, V2) and touching the circle ery? V2y- the mean of r, andr, Oare r, and r, respectively, then Column -1 Column = I (A) Number of common tangents of the circles 4 + y? — 2x =O and x? + y? + 6x - by +2 = Ois (8) Number of indirect common tangents of the circles @ 2 x4 y= Ax = 10y +4 = 08 X? + y?~ 6x — 12y - 55 = Dis (C) Number of common tangents of the circles x? + y?~ 2x - 4y = 0 © 3 Bx +y'—By-4=0is (0) Number of direct common tangents of the circles () 0 xi + y2+ 2x — By + 13 = 08 x? +? 6x 2y +6 =O is Scanned with CamScannerPART - ONLY ONE OPTION CORRECT TYPE » Marked Questions may have for Revision Questions, ton Sam Bam Sem. 4 1 1 1 to (aya (8) 16 (cy (0)2 From the point A (0: 3) on the circle x? + 4x + (y~3)? = 0 a chord AB is drawn & extended to a point M such that AM = 2 AB, The equation of the locus of M is: (A) + 8x+ y? = 0 (B) x + 8x + (y-3)7=0 (C) (K-3)* + 8x4y2=0 (D) x? + 8x + 8y? ftangent at (1, 2) to the circle c,: x? + y2= 5 intersects the circle c,: x? + y?= 9 at A & Band tangents atA& B to the second circle meet at point C, then the co-ordinates of C is 9 18 9 (A) (4, 5) (B) (22) (C) (4, -5) (0) ( 7 A circle passes through point (3 8 touches the line pair x?— y? - 2x + 1 = 0. Centre of circle lies inside the circle x? + y?~ 8x + 10y + 15 = 0. Co-ordinate of centre of circle is (A) (4, 0) (B) (5, 0) (C) (6, 0) (D) (0, 4) The length of the tangents from any point on the circle 15x? + 15y?— 48x + 64y = 0 to the two circles 5x? + Sy? — 24x + 32y + 75 = 0 and 5x? + 5y? ~ 48x + 64y + 300 = O are in the ratio (A) 1:2 (B)2:3 (C)3:4 (D)2:4 The distance between the chords of contact of tangents to the circle; x? + y?+ 2gx+ 2fy + ¢= 0 from the origin & the point (g°f) is: P+f-o P+ fc 3 [Fs Pe gefee etePee A) \atot ow oom iat If from any point P on the circle x* + y* + 2gx+ 2fy + ¢ = 0, tangents are drawn to the circle x84 y+ 2gx-+ fy + csin’a + (g?+ F) costa = 0: then the angle between the tangents is: Aa ()20 og (5 The locus of the mid points of the chords of the circle x? + y* + 4x —6y-12 = O which subtend an angle of 3 radians at its circumference is: (A) (x~2)? + (y + 3)? = 6.25 (B) (x + 2)? + (y~3)? = 6.25 (C) (x + 2) + (y- 3)? = 18.75, (D) (x + 2)? + (y + 3)? = 18.75 I the two circles, x? + y2+2 g,x +2 fy =0 & x? + y? + 2.g,x +2 fy = 0 touch each other then: Ls (A) f, 95 = fede (B) 3% (C) fife 192 (0) f, += 91+ 92 Scanned with CamScannerCircle 40.8 to 12. 13.0 14m, 15. 16. WN. Keircle touches a straight line (x + my +n = 0 & cuts the circle x" + y*= 9 orthogonally. The locus of centres of such circles is: (A) (0x my + ny? = (2 +m?) (x? + y?- 9) (B) (@x-+ my—n}* = ( (C) (¢x + my +n)? = (+m?) (8 +y? + 9) (D) ((x+ my=n)? = (2+ emi) 024-9) 3) (2 +y*= 9) | angles is The locus of the point at which two given unequal circles subtend equal (D) an ellipse (A) a straight line (B) acircle (C)aparabola Acircle is given by x? + (y =1)*= 1. Another circle G touches it externally and also the ¥-2%i8, then the locus of its centre is (A) {0x y) 7 = ay} U (0x yD y's 0) (BAY)? +0 (C) {ox y) x7 = YJ UO y) = < 0} (0) (x, y) 2 = Ay} U (Oy) 2 SOF internally and tangent on which y — 17 = 4) U (KY SO} ‘The locus of the centre of a circle touching the circle x* + y? — 4y ~ 2x= from (1, 2) is making a 60° angle with each other. (A) (X= 17 + (y= 2) (C) (x + 1 + (y-2) (B) (x12 + (y= 2 =4 (D) (x+ 1)? + (y+ 2h =4 three circles which are such that their centres are non-collinear, then exactly one circle STATEMENT: exists which cuts the three circles orthogonally. STATEMENT-2 : Radical axis for two intersecting circles is the common chord. (A) STATEMENT-1 Is true, STATEMENT-2 is true and STATEMENT-2 is correct explanation for STATEMENT-1 (®)__ STATEMENT-1 is true, STATEMENT.2 is true and STATEMENT-2 is not correct explanation for STATEMENT-1 (©) STATEMENT-1 is true, STATEMENT-2s false (0) STATEMENT-1 is false, STATEMENT-2is true The centre of family of circles cutting the family of 3 circles x? + y? + ax(2-§) + av(+-4) —6 (i + 2) = 0 orthogonally, lies on (A)x-y-1=0 (B)4x+3y-6=0 (C)4x+3y+7=0 — (D) 3x-4y~ The circle x + y? = 4 cuts the circle x? +y?+2x+ 3y-5=0 inA& B. Then the equation of the circle on AB as a diameter is: (A) 13(x7 +2) - 4x - 6y-50=0 (B) 9(x* + y?) + 8x - dy + 25=0 (C) x? + y? — 5x + 2y +72 (D) 13(x2 + y*)~ 4x- 6y+50=0 tam 38 SINGLE AND DOUBLE VALUE INTEGER TYPE PART - Il Find maximum number of points having integer coordinates (both x, y integer) which can lie on acircle with centre at (2, 4/3) is (are) The axes are translated so that the new equation of the circle x? + y*~5x+2y-5 = 0 has no first degree terms and the new equation x? + y?= a then find the value of 4, Find the sum of co-ordinates of the centre of the smallest circle touching the circles x2 + y#—2y—3 = O and x? + y?~ 8x - 18y + 93 = 0, Scanned with CamScannerCircle a 6. Tos 8x 12. 1328 14, 15. 16.5, N- Aline meets the co-ordinate axes in A and B. A circle is circumscribed about the triangle OAB, If 4, and d, are the distances of the tangent to the circle at the origin O from the points Aand B respectively and diameter of the circle is ,d, + 2,d,, then find the value of hy + hy Acircle is inscribed (i.e. touches all four sides ) into a rhombous ABCD with one angle 60°, The distance from the centre of the circle to the nearest vertex is equal to 1. If P is any point of the circle, then |PA|?+|PB)°+|PC|"+|PD)? is equal to: Find the number of integral points which lie on or inside the circle x? + ys4, Let x &y be the real numbers satisfying the equation x? ~ 4x + y?+ 3 = 0. If the maximum and minimum values of x* + y? are M & m respectively, then find the numerical value of (M + m). Find number of values of 'c' for which the set, {0% y) Ix? + y? + 2x5 1) 9 {(x, y) Ixy +6 2 0} contains only one point is common. A thombus is inscribed in the region common to the two circles x? + y? - 4x - 12 = 0 and x? + y* + 4x— 12 = O with two of its vertices on the line joining the centres of the circles and the area of the rhombus is ay/3 sq. units, then find the value of a. If (a, B) is a point on the circle whose centre is on the x-axis and which touches the line x + y = O at (2,-2), then find the greatest integral value of ‘a’ ‘Two circles whose radii are equal to 4 and 8 intersect at right angles. The length of their common cherd a YB v then find % 7 is Avariable circle passes through the point A (a, b) & touches the x-axis and the locus of the other end of the diameter through Ais (x ~ a)? = Aby , then find the value of i Let A be the centre of the circle x* + y* - 2x ~ 4y — 20 = 0. Suppose that the tangents at the points B (1,7) & D(4,~ 2) on the circle meet at the point C. Find the area of the quadrilateral ABCD. Find the greatest integer values of a for which the point (2a, a + 1) is an interior point of the larger segment of the circle x? + y? - 2x ~ 2y ~ 8 = 0 made by the chord whose equation is x-y + 1=0. The circles x? + y? + 2ax + cy +a=0 and x? + y?—3ax + dy then find the number of values ofa’ for which the line 5x + by — intersect in two distinct points P and Q, O passes through P and Q. The circumference of the circle x? + y? - 2x + 8y -q= O is bisected by the circle x2 + y24 4x + 12y + p= 0, then find p+q PART - Ill : ONE OR MORE THAN ONE OPTIONS CORRECT TYPE 3a The equation of circles passing through (3, -6) touching both the axes is (A) x2 + y?- 6x + 6y +9=0 (B) x2 + y? + 6x-6y + 9=0 (C) x2 + y2 + 30x - 30y + 225 = 0 (D) x? + y? ~ 30x + 30y + 225 = 0 Equations of circles which pass through the points (1, -2) and (3, - 4) and touch the x-axis is (A) x2 + y2+ 6x + 2y + (B) x2 + y? + 10x + 20y + 25 (C) x+y? 6x + 4y+9=0 (D) x? + y? + 40x + 20y ~ 25 The centre of a circle passing through the points (0, 0), (1, 0) & touching the circle x? + y2= 9 is : (a) @ 3) (8) (3.8) (0) (G8 Scanned with CamScanner5am 10 Mee 12. 1323. x, ve \ 9 % + X= 1 and lies in the “The equation of the circle which touches oth the axes andthe line 3 4 a quadrant is (x - ¢)? + (y ~¢)* = ©? where cS (6 (A)t (B)2 on the intersect jh ht fines which pass throug) a morthe circle +" = 100 into tw ion of the lines x ~ 2y~5=9 ‘arcs whose lengths are in the Find the equations of st 7x + y = 50 & divide the circumferenc ratio 2:1 (A) 3x—4y-25=0 (B) 4x + gy-25=0 " (8 These tanger ‘Tangents are drawn to the circle x + y (0 from a point 'P’ lying on ee oP eM ae y-axis at points P,and'P,’ Possible coordinates of 'P' so that area of va | isfare (0) 10 42.9) (A)(10, 0) (B) (10 J2.9) (C) (-10, 0) ~ax-a?= Ohas 2 = 4, then x? If (a, 0) is @ point on a diameter segment ofthe circle x? += 4, (A) exactly one real root in (= 1, 0} (@) Exactly one real root fe 5] (C) distinct roots greater than-1 (0) Distinct roots less than =o (0) 3x+4y-25=0 O are perpendicular if yas yt 2rx= 2hy + h?= (pear The tangents drawn from the origin to the circle (ortnet (ayner (ener gent atthe point(0, 0)tothe circle where circle makes intercepts of length 2aand ‘The equation (s) of the tan: 2b units on the coordinate axes, is (are) - . (A) ax + by =0 (B) ax-by=0 (C)x=y (D) bx + ay=ab Consider two circles C, :x?+ y2-1=Oand C,:x2+ y'=2= 0. Let A(1,0) be a fixed point on the circle C, I the circle C,, The line BA meets the curve C, again at c and B be any variable point of Which of the following alternative(s) is/are correct ? (A) OA? + OB? + BC? e [7, 11], where O is the origin. (B) OA? + OB? + BC? € [4, 7], where O isthe origin. (C) Locus of midpoint of AB is a circle of radius ¥ (0) Locus of midpoint of ABs acicle of area a One of the diameter ofthe circle circumscribing the rectangle ABCD is x~ Sy + 1 = 0. it oe of rectangle are the points (~2, 5) and (6, 5) respectively, then which of the following holds) 90 (A) Area of rectangle ABCD is 64 square units (B) Centre of circle is (2, 1) (C) The other two vertices of the rectangle are (~2, ~3) 5 (©) Equation of sides are x=-2, y=—3,x=5 and Scee a Three concentric circles of which the biggest is x? + y? matric €) . y?= 1, have th ine y =x* 1 cece the chcles in real and distinct points. The permissible values 6 rain. the era ou alues of common difference of AP (A)0.4 (8) 06 (0.01 oe Ife Sm? + 6¢ + 1 = 0, Prove that ¢x+ my +1 = : y+t1=0 , ened touches a definite circle, then which of the follow"? (A) Centre (0, 3) (B) centre (3, 0) . (C) Radius /5 a cia (0) Radius 5 —1 Scanned with CamScannerCircle 148 15. 16.0 7. 18. 19.0, 20. IN- Ifthe circle C,: x*+ y? = 16 intersects another circle C, of radius § in such a manner that the common chord is of maximum length and has a slope equal to 3/4, then the co-ordinates of the centre of C, are: 062) a2) o(2) Wal?—bm?+2dC+1=0, where a, b, dare fixed real numbers such that a + b= d?, then the line (x + my +1 = 0 touches a fixed circle (A) which cuts the x-axis orthogonally (B) with radius equal to b (C) on which the length of the tangent from the origin is fa?—b (D) none of these For the circles x + y?~ 10x + 16y + 89-1 = 0 and x? + y? + 6x — 14y + 42 = 0 which of the following islare true. (A) Number of integral values of rare 7 for which circles are intersecting. (B) Number of integral values of r are 9 for which circles are intersecting, (C) For r equal to 13 number of common tangents are 3. (0) For r equal to 21 number of common tangents are 2. Which of the following statement(s) is/are correct with respect to the circles S, = and S, =x? + y?—2x-4y+4=0? (A)S, and S, intersect at an angle of 90°, e+ y2-4=0 6 8 (8) The point of intersection of the two circle are (2, 0) and é §), 4 (©) Length of the common of chord of S, and S, is 75° (D) The point (2, 3) lies outside the circles S, and S,. x? + y®= a? and (x — 2a)? + y* = a? are two equal circles touching each other. Find the equation of circle (or circles) of the same radius touching both the circles. (A) x + y? + 2ax + 2/3 ay + 3a? =0 (B) xt + — 2ax + 2/3 ay + 3a?=0 (C)x+ y+ 2ax-2 JF ay + 3a*=0 (0) x + y*-2ax-2 Jay + 3a?=0 The circle x? + y? - 2x - 3ky - 2 = 0 passes through two fixed points, (kis the parameter) (4) (1+ 43.0) (8) (4+¥8,0) (c) (3-10) (0) (-v3.0) Curves ax? + 2hxy + by? ~ 29x — 2fy + ¢ = O and a’x?- 2hxy + (a' + a— by? 2g'x-2f'y +c=0 e+e +e a'ta’a'ta intersect at four concyclic pointA, B, C and D. If Pis the point ( ) «then which of the following islare true : (A) P is also concyclic with points A,B,C,D {B) PA, PB, PC in GP. (C) PA? + PB? + PC? = 3PD? (D) PA, PB, PC in AP. PART - IV : COMPREHENSION Comprehension # 1 (Q. No. 1 to 3). Let S,, S,, S, be the circles x? + y? + 3x + 2y+1=0, x? +y?—x+6y+5=0 and x+y? + 8x = By + 15 =0, then Scanned with CamScannerCircle ca 2m Point from which length of tangents to these three circles is same 'S (ent (A)(1..0) (B) (3, 2) (c) (10, §) Equation of circle S, which cut orthogonally to all given citcle is (A) x24 y?-6x 4 4y—14=0 (B)X +4 OF ng (C) + y?-6x-4y 4 14=0 {D) x + - x= 4¥ Radical centre of circles S,, S,, & S, is nS 38 (D) ( a! ) w(-3-3) ®)@.2) (0)(1,0) Ole 2 Comprehension # 2 (Q. No. 4 to 6) Two circles are 48 5s with centres C, & Adirect common tangent is drawn from a point P which touches S, &S, at Q&R, respectively. Find the ratio of area of APQC, & APRC,. (Ay3:4 (8) 9:16 (c) 16:9 Ole From point ‘A’ on S, which is nearest to C,, a variable chord is drawn to S,. The locus of mid point ofthe chord, (A) circle (B)Diameterofs, (C)Arcofacircle _—_(D) chord of s, butnot diameter Locus of 5 cuts the circle S, at B & C, then line segment BC subtends an angle on the major arc of circle S, is att 43 5 tant 4 z 13 % oot 4 (A) cos | (8) 5 - tar 5 (C)F-ztawtZ (DO) Foot" s PART - | : JEE (ADVANCED) / IIT-JEE PROBLEMS (PREVIOUS YEARS) * Marked Que: ions may have more than one correct option, ‘+3 Marked Questions may have for Revision Questions. 4. Tangents are dravun from the point (17, 7) tothe circle x? + y? = 169, NT-JEE - 2 A, (3-1), 162 STATEMENT-1: The tangents are mutually perpendicular. ' or bapets raise! because STATEMENT-2 : The locus of the points from which mutually perpendi Shon eis wate oF g38. ly perpendicular tangents can be drawn tothe Statement-1 is True, Statement-2 is True ; Statement-2 Is aco; ; Te (B) Statement-1 is True, Statement-2 is True ; Statement-2 Is NOT a ea er So tet (C) Statement-1 is True, Statement-2 is False explanation for Statement (0) Statement-1 is False, Statement-2 is True v& Let ABCD be a quadrilateral with area 18, with side AB oe Parallel to the side C| = be petpedeuler to AB and CD. Ia circle is drawn inside the quadrilateral ABCD roan to is UT-JEE- 2007, Paper-2, (3,- 1), 12) 3 of @2 2 “ (0)1 [et ASR me? Peace Ofes: 6 Tone, Rap a Scanned with CamScannerCircle mm 3. Leta and b be non-zero real numbers. Then, the the equation (ax? + by? +c) (x? —Sxy + 6) =O repre (A) four stelght ines, when ¢=0anda, bare fhe samesign fITIEE - 2008, Papert ay 22) AB) twosstraight lines and a circle, when a=, and cis of sign opposite tothatofa (C) two straight lines and a hyperbola, when a and b are of the same sign and cis of sign opposite to (0) _acircle and an ellipse, when a and b are of the same sign and cis of sign opposite to that of a = Astraight line through the vertex P of a triangle POR intersects the side QR at the point S a int S and the circum- Circle of the triangle POR atthe point TIS 5 no the centre ofthe crcurnice, then acm INT-JEE - 2008, Paper-2, (4, 0), 82] 104 2 44 2 ii. 2— A,i,72— () ps * ST * Jasxsr (Fes * St” JasxEr Omte 0, then the value of [K] is [IT-JEE 1 [Note : [k] denotes the largest integer less than or ‘equal to k] sat (0, 2) also passes through the poirt ‘The circle passing through the point (~1, 0) and touching the yay sa ee aort,Paper2 8-1 80 (-$. 2) ol 3 3) (D(-4,0) 1 region x? + y? <6 into two pars. (.3,) @(->9) ®) The straight line 2x — 3y = 1 divides the circular = (23)(8.23)(4- 2.4} ee oy a its= tae ( [UT-JEE 2011, Paper-2, (4, 0), 80] then the number of point(s) in $ lying inside the smaller part is {ts drawn from points lying on the straight line [IIT-JEE 2012, Paper-1, (3, -1), 70] (B) 20(x? + y2) + 36x — 45y (0) 36(¢e + y’) + 20x ~ 45y = ‘The locus of the mid-point of the chord of contact of tangent 4x — 5y = 20 to the circle x? + y? = 9 is (A) 20(x + y*) ~ 36x + 45y = 0 (C) 36(x? + y)— 20x + 45y = 0 Paragraph for Question Nos. 15 to 16 ‘Atangent PT is drawn to the circle x? + y°= 4 atthe point P(_J3 , 1).Astraight ine L, perpendicular to PT is a tangent to the circle (x - 3)? + y°= 1 [IIT-JEE 2012, Paper-2, (3,1), 66] ‘A.common tangent of the two circles is (Ayxa4 (B)y=2 (C)x+ By=4 ()x+2 /ay=6 A possible equation of L is (ayx- Jaye (8)x+ By (C)x- By= (O)x+ JSy=5 Circles) touching x-axis at a distance 3 from the origin and having an intercept of length 2/7 isis = 2y7 ony-axi [JEE W ¥4y'orr8/¢9=0 (0) x ogre eargre) 2013, Paper 2, (3, A¥60] (©) + y°-6x- By +9=0 (0) €+y'~6x-7y49=0 force ; passes through the point (0, 1) and is orthogonal to the circles ( id eye x = 1) + yF= 16 ani i [JEE (Adi (A) radius of Sis 8 LEE (Advanced) 2014, Paper-t, (3, 0)/60 GeanvestSisc7, 1 yee ee, eon -8, 1) From Of G6 Toner AEE Resonance’ SW Rai Ka a SE Scanned with CamScannerCircle WN PART - Il : JEE (MAIN) / AIEEE PROBLEMS (PREVIOUS YEARS) 10. 1. 12, 13. Ifthe lines 3x—4y ~7 = 0 and 2x — ay —5 = etesation’of the eecis ie O are two diameters of a circle of area 49n square units, the (1) x + y+ 2x 2y-62=0 hee IAIEEE 2006, (3, ~1), 120] (3) x? +? = 2x + 2y—47=0 (2)x? + 2% + ay ~6: Let C be the circl ee ye the ci et ircle with centre (0, 0) and radius 3 units. The equation of the locus of the mid points of the chords of the circle C that subtend an angle of 2% 3 at its centre, is [AIEEE 2006, (3, -1), 120] — avery et veyed ( axrys 2 @wryso toe sy= 3 Consider a family of circles which are passing through the point (1,1 (3 i are the coordinates of the centre of the circles, then the set of values ce gran ts Renilla of kis given by the interval IAIEEE 2007, (3,~1), 120] (3)-12 0) touch each other if : [AIEEE-2011, |, (4,—1), 120] (1) 2lal=e (2) lal=c (3) a= 2c (4) lal = 2 The equation of the circle passing through the point (1, 0) and (0, 1) and having the smallest radius is - (1) x? + y= 2x-2y + 1=0 (2)x¢+y@—x-y=0 [AIEEE-2014, I, (4,~1), 120] (3) x2 +y?+ 2x + 2y-7=0 (4) xt 4 ytexty-2=0 The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the Point (2, 3) is [AIEEE- 2012, (4,—1), 120] 10 3 6 5 oz Qs Os Ms The circle passing through (1, -2) and touching the axis of x at (3, 0) also passes through the point [AIEEE - 2013, (4,~1),120] (1) (-5, 2) (2) (2,-5) (3) (5-2) (4) (-2, 5) Let be the circle with centre at (1, 1) and radius = 1. If Tis the circle centred at (0, y), passing through origin and touching the circle C externally, then the radius of Tis equal to [JEE(Main) 2014, (4,—1), 120] 3 1 1 a > Ay 8p as it (2, 3) in the line (2x - 3y + 4) + k (x-2y+3)=0,keR,isa Locus of the image of the point (2, 3) TEE(Main) 2016, (4,=1), 120] (1) straight line parallel to x-axis (2) straight line parallel to y-axis (3) circle of radius /2 (4) circle of radius JS circles x? + y? — 4x By — 12 = O and x? + y?+ 6x + 18y + 26 = 0, is [JEE(Main) 2015, (4,—1), 120] (4) 4 The number of common tangents to the (yt (2)2 )3 Scanned with CamScannerWN Cirele Section (E) EXERCISE - 1 Baer Ed. x@ty?-17x- 19y +50=0 Section (A) E2 x+y?-2x-4y=0. At xtyeet PART-II AS. 43x 4y=0 Section (A) Ad xttylax—dy +420 Ad (0) AZ (A) AS (8) AS. xi+y?t6 V2ye6x+9=0 A4. (Cc) AS. (0) AB (8) AG. (x+3P +(y-4)?=4 AT. (36,47) AT. (A) AB (C) Section (B) Section (B) BA. (1,3),6,7), 442 B4. (A) B2 (8) BS 8)" B22 x-7y-45=0 B4, (B) BS. (A) B6 (B) B3. \3x-y24=0 B7. (A) BS (D0) B89 (C) Bao, (A) B41. (C) B42 (A) B4, 16x? 65y?— 288x + 1296 = 0, tan* G4) B-13. (B) B14. (A) BS. Yes Section (C) Section (C) C4. (A) C2 (B) C3. (A) C4. 2x-y=0 C4. (C) C5. (B) Ce. (A) C2. x+2y-1=0 cz. (A) C3. (x+4)+y?=16 Section (D) ca (« 7 8) D4. (8) D2 (8) Ds (A) D4. (A) C6. xty+5=0 Section (E) Section (D) ‘ Ei (A) EZ (A) ES. (A) A. X= 0, 3x + dy = 10, y= 4 and dy = 4x, Ye sera ieee PART-II D3, (x? +y?)- 7x + 2y=0 x 4 A.B >.) 51, (6) =,2 i ~~ & ) 2A) (0. (8) > (6), (C) >), (0) 4) Resenence Scanned with CamScannereee mN- EXERCISE - 2 PARTE PART -I 1 ® 2 ©) 3 &W . © 2% @ 3 Ay (BY) Sete (Cs B11) 4 A & AY By EXERCISE - 3 nr ©) ®& (8) 9 (B) PART-1 . A) HB) 12) 1 WwW 2 & 2 & 3. «©(8)) 14) 15. (B) 4. (BD) 5 oO 6 (A) 46. (A) 7. 2 8 © »% ®) PART-II 10. «8 3 12, (0) 4 1 2. 49 3 D 13. 2 14. (A) rr) 4 2 5. t 6 ‘a 46. (A) 17" (AC) 18." (BC) er eelo ee ames ees PART -Il Plreenare (2) eee 2. ueeees (3) eran Seema) 4 @ & (@) 6& (1) 3°75 141 18 . 2® & @ ® 16. 10 % -.@) @ 2. @ % @ 2 @ PART - III 2% @ 1 (aD) 2) (CB) BD) 4. (AD) 5 = (CD) & (AC) 7. (ABCD) 8. (ABD) 9 —(AB) 10. (ACD) 11. (ABC) 12. (CD) 13. (BC) 14, (BD) 15. (AC) 16. (AC) 17. ~~ (ACD) 18. ~— (8D) 19. (AD) 20. (BCD) Scanned with CamScanner