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JEE Main Circles questions

57 Mathematics questions on Circles as asked in JEE Main, newest first. 34 come with a worked solution.

  1. The locus of the mid points of the chords of the circle C_1:(x-4)^2+(y-5)^2=4 which subtend an angle θ_i at the centre of the circle C_1, is a circle of radius r_i. If θ_1=(π)/3, θ_3=(2π)/3 and r_1^2=r_2^2+r_3^2, then…

    2023

  2. The points of intersection of the line ax + by = 0, (a ≠ b) and the circle x^2 + y^2 - 2x = 0 are A(α, 0) and B(1, β). The image of the circle with AB as a diameter in the line x + y + 2 = 0 is :

    2023

  3. Let the tangents at the points A(4,-11) and B(8,-5) on the circle x^2 + y^2 - 3x + 10y - 15 = 0, intersect at the point C. Then the radius of the circle, whose centre is C and the line joining A and B is its tangent, is…

    2023

  4. Points P(-3, 2), Q(9, 10) and R(a, 4) lie on a circle C with PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line 2x - ky = 1, then k is equal to

    2023

  5. If the circles x^2 + y^2 + 5Kx + 2y + K = 0 and 2(x^2 + y^2) + 2Kx + 3y - 1 = 0, (K ∈ R), intersect at the points P and Q, then the line 4x + 5y - K = 0 passes through P and Q, for

    2019 · with solution

  6. The locus of the centres of the circles, which touch the circle, x^2 + y^2 = 1 externally, also touch the Y-axis and lie in the first quadrant, is

    2019

  7. If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is

    2019 · with solution

  8. The common tangent to the circles x^2 + y^2 = 4 and x^2 + y^2 + 6x + 8y - 24 = 0 also passes through the point

    2019

  9. A circle touching the X-axis at (3, 0) and making a intercept of length 8 on the Y-axis passes through the point

    2019

  10. If a tangent to the circle x^2 + y^2 = 1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is

    2019 · with solution

  11. The sum of the squares of the lengths of the chords intercepted on the circle, x^2 + y^2 = 16, by the lines, x + y = n, n ∈ N, where N is the set of all natural numbers, is

    2019 · with solution

  12. A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (-8, 5) and (6, 5), then the area of the rectangle (in sq units) is

    2019 · with solution

  13. The line x = y touches a circle at the point (1, 1). If the circle also passes through the point (1, -3), then its radius is

    2019 · with solution

  14. A circle cuts a chord of length 4a on the X-axis and passes through a point on the Y-axis, distant 2b from the origin. Then, the locus of the centre of this circle, is

    2019 · with solution

  15. If a circle of radius R passes through the origin O and intersects the coordinate axes at A and B, then the locus of the foot of perpendicular from O on AB is

    2019

  16. The tangent and the normal lines at the point ( √(3), 1) to the circle x^2 + y^2 = 4 and the X-axis form a triangle. The area of this triangle (in square units) is

    2019 · with solution

  17. Two circles with equal radii are intersecting at the points (0, 1) and (0, -1). The tangent at the point (0, 1) to one of the circles passes through the centre of the other circle. Then, the distance between the centres…

    2019 · with solution

  18. If a variable line, 3x + 4y - λ = 0 is such that the two circles x^2 + y^2 - 2x - 2y + 1 = 0 and x^2 + y^2 - 18x - 2y + 78 = 0 are on its opposite sides, then the set of all values of λ is the interval

    2019 · with solution

  19. Three circles of radii a, b, c (a < b < c) touch each other externally. If they have X-axis as a common tangent, then

    2019

  20. The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then, the sum of perpendicular distances from A and B on the tangent to the circle at the origin is

    2019

  21. Let C_1 and C_2 be the centres of the circles x^2 + y^2 - 2x - 2y - 2 = 0 and x^2 + y^2 - 6x - 6y + 14 = 0 respectively. If P and Q are the points of intersection of these circles, then the area (in sq units) of the…

    2019 · with solution

  22. A square is inscribed in the circle x^2 + y^2 - 6x + 8y - 103 = 0 with its sides parallel to the coordinate axes. Then, the distance of the vertex of this square which is nearest to the origin is

    2019

  23. If a circle C passing through the point (4, 0) touches the circle x^2 + y^2 + 4x - 6y = 12 externally at the point (1, -1), then the radius of C is

    2019

  24. If the circles x^2 + y^2 - 16x - 20y + 164 = r^2 and (x-4)^2 + (y-7)^2 = 36 intersect at two distinct points, then

    2019 · with solution

  25. If the area of an equilateral triangle inscribed in the circle, x^2 + y^2 + 10x + 12y + c = 0 is 27√(3) sq units, then c is equal to

    2019

  26. If the tangent at (1, 7) to the curve x^2 = y - 6 touches the circle x^2 + y^2 + 16x + 12y + c = 0, then the value of c is

    2018 · with solution

  27. The radius of a circle having minimum area, which touches the curve y = 4 - x^2 and the lines y = |x|, is

    2017

  28. The centres of those circles which touch the circle, x^2 + y^2 - 8x - 8y - 4 = 0, externally and also touch the X-axis, lie on

    2016 · with solution

  29. If one of the diameters of the circle, given by the equation, x^2 + y^2 - 4x + 6y - 12 = 0, is a chord of a circle S, whose centre is at (-3, 2), then the radius of S is

    2016 · with solution

  30. The number of common tangents to the circles x^2 + y^2 - 4x - 6y - 12 = 0 and x^2 + y^2 + 6x + 18y + 26 = 0 is

    2015 · with solution

  31. Let C be the circle with centre at (1, 1) and radius 1. If T is the circle centred at (0, y) passing through origin and touching the circle C externally, then the radius of T is equal to

    2014 · with solution

  32. The circle passing through (1, -2) and touching the axis of x at (3, 0) also passes through the point

    2013 · with solution

  33. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle x^2 + y^2 = 9 is

    2012 · with solution

  34. The circle passing through the point (-1, 0) and touching the Y-axis at (0, 2) also passes through the point

    2011 · with solution

  35. Tangents drawn from the point P(1, 8) to the circle x^2 + y^2 - 6x - 4y - 11 = 0 touch the circle at the points A and B. The equation of the circumcircle of triangle PAB is

    2009

  36. Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its…

    2007 · with solution

  37. The locus of the centre of circle which touches (y-1)^2 + x^2 = 1 externally and also touches X-axis, is

    2005 · with solution

  38. The tangent at (1, 7) to the curves x^2 = y - 6x touches the circle x^2 + y^2 + 16x + 12y + c = 0 at

    2005 · with solution

  39. If one of the diameters of the circle x^2 + y^2 - 2x - 6y + 6 = 0 is a chord to the circle with centre (2, 1), then the radius of the circle is

    2004

  40. The centre of circle inscribed in square formed by the lines x^2 - 8x + 12 = 0 and y^2 - 14y + 45 = 0, is

    2003 · with solution

  41. If the tangent at the point P on the circle x^2 + y^2 + 6x + 6y = 2 meets the straight line 5x - 2y + 6 = 0 at a point Q on the Y-axis, then the length of PQ is

    2002 · with solution

  42. Let AB be a chord of the circle x^2 + y^2 = r^2 subtending a right angle at the centre. Then, the locus of the centroid of the triangle PAB as P moves on the circle, is

    2001

  43. Let PQ and RS be tangents at the extremities of the diameter PR of a circle of radius r. If PS and RQ intersect at a point X on the circumference of the circle, then 2r equals

    2001 · with solution

  44. If the circle x^2 + y^2 + 2x + 2ky + 6 = 0 and x^2 + y^2 + 2ky + k = 0 intersect orthogonally, then k is

    2000 · with solution

  45. The triangle PQR is inscribed in the circle x^2 + y^2 = 25. If Q and R have coordinates (3, 4) and (-4, 3) respectively, then ∠ QPR is equal to

    2000 · with solution

  46. If two distinct chords, drawn from the point (p, q) on the circle x^2 + y^2 = px + qy (where, pq ≠ 0) are bisected by the X-axis, then

    1999 · with solution

  47. The number of common tangents to the circles x^2 + y^2 = 4 and x^2 + y^2 - 6x - 8y = 24 is

    1998

  48. The locus of the centre of a circle, which touches externally the circle x^2 + y^2 - 6x - 6y + 14 = 0 and also touches the Y-axis, is given by the equation

    1993

  49. The centre of a circle passing through the points (0, 0), (1, 0) and touching the circle x^2 + y^2 = 9 is

    1992 · with solution

  50. The lines 2x - 3y = 5 and 3x - 4y = 7 are diameters of a circle of area 154 sq units. Then, the equation of this circle is

    1989

  51. If the two circles (x-1)^2 + (y-3)^2 = r^2 and x^2 + y^2 - 8x + 2y + 8 = 0 intersect in two distinct points, then

    1989 · with solution

  52. If a circle passes through the point (a, b) and cuts the circle x^2 + y^2 = k^2 orthogonally, then the equation of the locus of its centre is

    1988

  53. The locus of the mid-point of a chord of the circle x^2 + y^2 = 4 which subtends a right angle at the origin, is

    1984 · with solution

  54. AB is a diameter of a circle and C is any point on the circumference of the circle. Then,

    1983 · with solution

  55. The equation of the circle passing through (1, 1) and the points of intersection of x^2 + y^2 + 13x - 3y = 0 and 2x^2 + 2y^2 + 4x - 7y - 25 = 0 is

    1983

  56. The centre of the circle passing through the point (0, 1) and touching the curve y = x^2 at (2, 4) is

    1983

  57. Two circles x^2 + y^2 = 6 and x^2 + y^2 - 6x + 8 = 0 are given. Then the equation of the circle through their points of intersection and the point (1, 1) is

    1980 · with solution

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