Tiêu đề XI - maths - chapter 11 - CIRCLES-II.pdf ✅

26 NARAYANAGROUP CIRCLES JEE-MAIN SR-MATHS VOL-IV EQUATION OF CIRCLE, CENTRE-RADIUS: 1. The line x+y=1 cuts the coordinate axes at Pand Q and a line perpendicular to it meet the axes in R and S. The equation to the locus of the point of intersection of the lines PS and QR is 1) 2 2 x y  1 2) 2 2 x y x y     2 2 0 3) 2 2 x y x y     0 4) 2 2 x y x y     0 2. The abscissae of two points A and B are the roots of the equation 2 2 x ax b    2 0 and their ordinates are the roots of the equation 2 2 y py q    2 0 then the radius of the circle with AB as diameter is 1) 2 2 2 2 a b p q    2) 2 2 a p  3) 2 2 b q  4) 2 2 2 2 a b p q    3. A rod AB of length 4 units moves horizontally with its left end A always on the circle 2 2 x y x y      4 18 29 0 then the locus of the other end B is 1) 2 2 x y x y      12 8 3 0 2) 2 2 x y x y      12 18 3 0 3) 2 2 x y x y      4 8 29 0 4) 2 2 x y x y      4 16 19 0 4. A circle of constant radius 3k passes through (0 ,0) and cuts the axes in A and B then the locus of centroid of triangle OAB is 1) 2 2 2 x y k   2) 2 2 2 x y k   2 3) 2 2 2 x y k   3 4) 2 2 2 x y k   4 5. A rod PQ of length 2a slides with its ends on the axes. The locus of the circumcentre of OPQ is 1) 2 2 2 x y a   2 2) 2 2 2 x y a   4 3) 2 2 2 x y a   3 4) 2 2 2 x y a   6. A right angled issosceles triangle is inscribed in the circle 2 2 x y x y      4 2 4 0 then length of its side is 1) 2 2)2 2 3)3 2 4) 4 2 7. The locus of the foot of the perpendicular drawn from origin to a variable line passing through fixed point (2,3) is a circle whose diameter is 1) 13 2) 13 2 3) 2 13 4) 26 8. A square is inscribed in the circle 2 2 x y x y      2 8 8 0 whose diagonals are parallel to axes and a vertex in the first quadrant is A then OA is 1)1 2) 2 3) 2 2 4)3 9. The equation of the image of the circle 2 2 x y x y      6 4 12 0 by the mirror x+y-1=0 is 1) 2 2 x y x y      2 4 4 0 2) 2 2 x y x y      2 4 4 0 3) 2 2 x y x y      2 4 4 0 4) 2 2 x y x y      2 4 4 0 CIRCUMSCRIBING AND INSCRIBING CIRCLES, CONCYCLIC POINTS 10. The circle passing through (t , 1) , (1 , t) and (t , t) for all values of t also passes through 1) (0 , 0) 2) (1 , 1) 3) (1,-1) 4) (-1,-1) 11. ABCD is a square with side ‘a’. If AB and AD are taken as positive coordinate axes then equation of circle circumscribing the square is 1) 2 2 x y ax ay     0 2) 2 2 x y ax ay     0 3) 2 2 x y ax ay     0 4) 2 2 x y ax ay     0 12. Two rods of lengths ‘a’and ‘b’slide along coordinate axes such that their ends are concyclic.Locus of the centre of the circle is 1) 2 2 2 2 4( ) x y a b    2) 2 2 2 2 4( ) x y a b    3) 2 2 2 2 4( ) x y a b    4) xy ab  LEVEL-II - (C.W)
NARAYANAGROUP 27 JEE-MAIN SR-MATHS VOL-IV CIRCLES POWER OF A POINT, POSITION OF A POINT, CHORD 13. The locus of centre of a circle which passes through the origin and cuts off a length of 4 units on the line x=3 is (EAMCET-2009) 1) 2 y x   6 0 2) 2 y x   6 13 3) 2 y x   6 10 4) 2 x y   6 13 14. If a chord of circle 2 2 x y   8 makes equal intercepts of length ‘a’ on the coordinate axes then a  1) 2 2) 4 3) 2 2 4) 8 15. The triangle PQR is inscribed in the circle 2 2 x y   25 . If Q=(3,4) and R=(-4,3) then QPR = 1) 2  2) 3  3) 4  4) 6  TANGENTS, ANGLE BETWEEN TANGENTS, NORMAL, PAIR OF TANGENTS, LENGTH OF THE TANGENT 16. The locus of the point (l, m). If the line lx+my=1 touches the circle 2 2 2 x y a   is 1) 2 2 2 x y a   2 2) 2 2 2 2 2 x y a   3) 2 2 2 a x y ( ) 1   4) 2 2 2 a x y ( ) 2   17. Tangents AB and AC are drawn to the circle 2 2 x y x y      2 4 1 0 from A(0,1) then equation of circle passing through A,B and C is 1) 2 2 x y x y      2 0 2) 2 2 x y x y      2 0 3) 2 2 x y x y      2 0 4) 2 2 x y x y      2 0 18. Locus of the point of intersection of tangents to the circle 2 2 x y x y      2 4 1 0 which include an angle of 0 60 is 1) 2 2 x y x y      2 4 19 0 2) 2 2 x y x y      2 4 19 0 3) 2 2 x y x y      2 4 19 0 4) 2 2 x y x y      2 4 19 0 19. Locus of point of intersection of perpendicular tangents to the circle 2 2 x y x y      4 6 1 0 is 1) 2 2 x y x y      4 6 15 0 2) 2 2 x y x y      4 6 15 0 3) 2 2 x y x y      4 3 15 0 4) 2 2 x y x y      4 6 15 0 20. The condition that the pair of tangents drawn from origin to circle 2 2 x y gx fy c      2 2 0 may be at right angle is 1) 2 2 g f c   2) 2 2 g f c   2 3) 2 2 g f c    2 0 4) 2 2 g f c   2 21. Number of circles touching all the lines x y   1 0 , x-y-1=0 and y+1=0 are 1) 0 2) 2 3) 4 4) Infinite 22. Number of circles touching all the lines x-2y+1=0, 2x+y+3=0 and 4x-8y+3=0 is 1) 0 2) 2 3) 4 4) Infinite 23. If y = 3x is a tangent to a circle with centre (1,1) then the other tangent drawn through (0,0) to the circle is 1) 3y = x 2) y = -3x 3) y = 2x 4) 3y = -2x 24. If the line y=x touches the circle 2 2 x y gx fy c      2 2 0 at P where OP =6 2 then c= 1) 36 2) 72 3) 18 4) 144 25. Tangents to 2 2 2 x y a   having inclinations  and  intersect at P. If cot cot 0     then the locus of P is 1) x+y=0 2) x-y=0 3) xy=0 4) 2 xy a  CIRCLES TOUCHING AXES, INTERCEPTS ON AXES 26. Equation of circles which touch both the axes and also the line x = k (k>0) is 1) 2 2 2 0 4 k x y kx ky      2) 2 2 2 0 4 k x y kx ky      3) 2 2 2 0 4 k x y kx ky      4) 2 2 2 0 4 k x y kx ky     
28 NARAYANAGROUP CIRCLES JEE-MAIN SR-MATHS VOL-IV 27. If two circles touching both the axes are passing through (2, 3) then length of their common chord is 1) 2 2) 2 2 3) 3 2 4) 4 2 28. Consider a family of circles which are passing through the point (-1, 1) and are tangents to x- axis. If (h , k) are the coodinates of the centre of circles then set of values of 'k' is given by the interval (EAMCET-2007, AIEEE-2008) 1) 1 1 2    K 2) 1 2 K  3) 1 2 O K  4) 1 2 K  29. A variable circle passes through the fixed point (2, 0) and touches y-axis then the locus of its centre is 1) Circle 2) parabola 3) Ellipse 4) stright line 30. A circle passes through A (1,1) and touches x - axis then the locus of the other end of the diameter through 'A' is 1)   2 x y   1 4 2)   2 y x   1 4 3)   2 x y   1 4 4)   2 y x   1 4 31. Equations of circles which touch both the axes and whose centres are at a distance of 2 2 units from origin are 1) 2 2 x y x y      4 4 4 0 2) 2 2 x y x y      2 2 4 0 3) 2 2 x y x y      4 0 4) 2 2 x y    4 0 MID POINT OF CHORD 32. If the tangent at (3 ,-4) to the circle 2 2 x y x y      4 2 5 0 cuts the circle 2 2 x y x y      16 2 10 0in A and B then the midpoint of AB is 1) (-6, -7) 2) (2, -1) 3) (2, 1) 4) (5, 4) 33. The locus of midpoints of the chord of the circle 2 2 x y   25 which pass through a fixed point (4,6) is a circle .The radius of that circle is 1) 52 2) 2 3) 13 4) 10 34. Locus of mid points of chords to the circle 2 2 x y x y      8 6 20 0which are parallel to the line 3x+4y+5=0 is 1) 3x+4y-25=0 2) 4x+3y+5=0 3) 4x-3y-25=0 4) 4x-3y+25=0 35. Locus of midpoints of chords of circle 2 2 2 x y r   having a constant length ‘2l’ is 1) 2 2 2 2 x y l r    2) 2 2 2 2 x y r l    3) 2 2 2 x y l   4 4) 2 2 2 2 x y l r    36. Let C be the circle with centre (0,0) and radius 3 units.The equation of locus of midpoints of chords of the circle C that subtend an angle of 2 3  at its centre is 1) 2 2 3 2 x y   2) 2 2 x y  1 3) 2 2 27 4 x y   4) 2 2 9 4 x y   37. The parametric equations   2 2 2 1 1 a t x t    and 2 4 1 at y t   represents a circle whose radius is 1) a 2) 2a 3) 3a 4) 4a 38. If a straight line through C 8, 8 making an angle 0 135 with the x -axis and cuts the circle x y   5cos , 5sin   in points A and B then AB= 1) 5 2) 10 3) 25 4) 16 39. The locus of the point which divides the join of A(-1, 1) and a variable point P on the circle 2 2 x y   4 in the ratio 3 : 2 is 1)     2 2 25 20 28 0 x y x y      2)     2 2 25 20 28 0 x y x y      3)     2 2 25 20 28 0 x y x y      4)     2 2 25 20 28 0 x y x y      RELATIVE POSITIONS OF CIRCLES & COMMON TANGENTS 40. If the circles 2 2 x y   2 and 2 2 x y x y      4 4 0  have exactly three real common tangents then  = 1) -10 2) 6 3) -6 4) 10
NARAYANAGROUP 29 JEE-MAIN SR-MATHS VOL-IV CIRCLES 41. The common tangents to the circles 2 2 x y x    6 0 , 2 2 x y x    2 0 form 1) Right angled triangle 2) Isosceles triangle 3) Equilateral triangle 4) Isosceles right angled triangle 42. If 1 , 1 3         is a centre of similitude for the circles 2 2 x y  1 and 2 2 x y x y      2 6 6 0 , then the length of common tangent of the circles is 1) 1 3 2) 4 3 3)1 4) Cannot be determined 43. Locus of the centre of the circle which touches 2 2 x y x y      6 6 14 0 externally and also y-axis is 1) 2 y x y     6 10 14 0 2) 2 y x y     6 10 14 0 3) 2 y x y     6 10 14 0 4) 2 y y x     6 10 14 0 AREAS 44. A rectangle ABCD is inscribed in a circle with a diameter lying along the line 3y = x+10. If A = (-6, 7), B = (4,7) then area of the rectangle in sq. units is 1) 80 2) 40 3) 160 4) 20 45. The area of the triangle formed by the tangent drawn at the point (-12, 5) on the circle 2 2 x y  169 with the coordinate axes is 1) 625 24 2) 28561 120 3) 225 23 4) 8561 20 46. Let AB be the chord 4x-3y+5=0 of the circle 2 2 x y x y      2 4 20 0 .If C =(7, 1) then the area of triangle ABC is 1) 15 sq.uint 2) 20 sq.unit 3) 24 sq.unit 4) 45 sq.unit 47. The minimum distance between the circle 2 2 x y 9   and the curve 2 2 2x 10y 6xy 1    is 1) 2 2 2) 2 3) 3 2  4) 1 3 11  48. ABCD is square of unit area. A circle is tangent to two sides of ABCD and passes through cxactly one of its vertices. The radius of the circle is 1) 2 2  2) 2 1 3) 1 2 4) 1 2 49. If 2 2 2 2 x y 16,x y 36     are two circles and P and Q move respectively on these circles such that PQ=4 then the locus of mid point of PQ is a circle of radius is 1) 20 2) 22 3) 30 4) 32 50. Number of positions of P such that APB 90   and area of triangle is 5. Where A = (1,2), B = (1,6) 1) 0 2) 2 3) 4 4)  LEVEL-II (C.W.) - KEY 1) 3 2) 1 3) 2 4) 4 5) 4 6) 3 7) 1 8) 2 9) 2 10) 2 11) 1 12) 3 13) 2 14) 2 15) 3 16) 3 17) 2 18) 1 19) 1 20) 2 21) 3 22) 2 23) 1 24) 2 25) 3 26) 1 27) 1 28) 4 29) 2 30) 3 31) 1 32) 1 33) 3 34) 3 35) 2 36) 4 37) 2 38) 2 39) 4 40) 2 41) 3 42) 3 43) 4 44) 1 45) 2 46) 3 47) 2 48)1 49)2 50)1 LEVEL-II (C.W.) - HINTS 1) Since R is orthocentre QR is 3rd altitude. The circle on P,Q as diameter 2) adding two equations 3) A is h r k r   cos , sin    and B is (x,y) and eliminate 4) O A a B b (0,0), ( ,0), (0, ) and G(x,y) and AB k  6 5) same as above 6)   2 2 2 x x r   2 7) radius is 2 2 1 1 x y  since it is the midpoint 8) same as above 9) One fourth of the area since rays are perpendicular 10) t = 1 11) draw diragram & observe